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THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

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Then  there  was  Charles  Davies,  appointed  in  1857  to  be  Professor  of 
Higher  Mathematics.  Until  Davies  began  his  activity,  the  algebra  and 
geometry  that  were  tauo-ht  in  the  United  States  were  those  that  had  been 
taught  at  Oxford  for  200  years,  the  geometry  being  based  on  Todhunter's 
Euclid.  It  remained  for  Davies  to  bring  to  America  knowledge  of  the  new 
and  strikingly  original  ideas  of  the  French  mathematicians.  In  this  way 
he  gave  our  people  a  new  mode  of  approach  and  a  new  mode  of  thinking 
in  reference  to  mathematics.  The  textbooks  of  the  Frenchman,  Legendre, 
were  translated  and  edited  by  Davies  and  came  into  use  all  over  the  United 
States.  l-t>vM/»v''Vv-c*''*«.4^'  'f<^'; 


ELEMENTS 


OF 


ALGEBRA: 


INCLUDING    STURM  S'    THEOREM. 


TRANSLATED    FROM     THE    FRENCH    OF 


M.  BOURDON. 


ADAPTED    TO    THE    COURSE    OF    MATHEMATICAL    INSTRUCTION   IN   THB 

DNITED    STATES, 

BY  CHARLES  DAVIES,  LL.D. 

AUTHOR   OF   ARITHMETIC,    ELEMENTARY   ALGEBRA,   ELEMEKTARY   GEOMETRT, 
PRACTICAL   GEOMETRY,   ELEMENTS   OF   SURVEYING,   ELEMENTS   OF 
DESCRIPTIVE    AND    ANALYTICAL   GEOMETRY,   ELEMENTS 
OF   DIFFERENTIAL   AND    INTEGRAL   CALCULUS, 
AND    A   TREATISE   ON    SHADES,    SHAD- 
OWS, AND   PERSPECTIVE. 


NEW  YORK: 

PUBLISHED  BY  A.  S.  BARNES  &  CO. 

CINCINNATI:— H.  W.  DERBY  &  CO. 

1850. 


DAVIE  S' 

COURSE   OF  MATHEMATICS. 


DAVIES'  FIRST  LESSONS  IN  ARITHMETIC— For  Beginners. 

I)  A  VIES'  ARITHMETIC— Designed  for  the  use  of  Academies  and  Schools. 

KEY  TO  DAVIES'  ARITHMETIC. 

DAVIES'  UNIVERSITY  ARITIIMETIC—Embracing  the  Science  of  Num- 
bers and  their  numerous  Applications. 

KEY  TO  DAVIES'  UNIVERSITY  ARITHMETIC. 

DAVIES'  ELEMENTARY  ALGEBRA— Being  an  introduction  to  the  Sci- 
ence, and  forming  a  counectiug  link  between  Akithmetic  and  Algebra. 

KEY  TO  DAVIES'  ELEMENTARY  ALGEBRA. 

DAVIES'  ELEMENTARY  GEOMETRY.— This  work  embraces  the  ele- 
mentary principles  of  Geometiy.  The  reasoning  is  plain  and  concise,  but  at  the 
same  time  strictly  rigorous. 

DAVIES'  ELEMENTS  OF  DRAWING  AND  3IENSURATION  —  Ap- 
plied to  the  Mechanic  Arts. 

DAVIES'  BOURDON'S  ALGEBRA— Inchiding  Sturm's  Theorem— Being 

an  abridgment  of  the  Work  of  M.  Bourdon,  with  the  addition  of  practical  examples. 

DAVIES'  LEGENDRE'S  GEOMETRY  and  TRIGONOMETRY— Being 

an  abridgment  of  the  work  of  M.  Legendre,  with  the  addition  of  a  Treatise  on  Men- 
suration OF  Planes  and  Solids,  and  a  Table  of  Logarithms  and  Logarithmic 
Sines. 

DAVIES'  SURVEYING— With  a  description  and  plates  of  the  Theodolite, 

Compass,  Plane-Table,  and  Level;  also,  Maps  of  the  Topographical  Signs 
adopted  by  the  Engineer  Department — an  explanation  of  the  method  of  surveying 
the  Public  Lands,  and  an  Elementary  Treatise  on  Navigation. 

DAVIES'  ANALYTICAL  GEOMETRY  — Embracing  the  Equations  of 
the  Point  and  Straight  Line — of  the  Conic  Sections — of  the  Line  and  Plank 
IN  Space  ;  also,  the  discussion  of  the  General  Equation  of  the  second  degree,  and 
of  Surfaces  of  the  second  order. 

DAVIES'  DESCRIPTIVE  GE03IETRY— With  its  application  to  Spher- 
ical Projections. 

DAVIES'  SHADOWS  and  LINEAR  PERSPECTIVE. 

DAVIES'  DIFFERENTIAL  and  INTEGRAL  CALCULUS. 


Entered,  according  to  Act  of  Congress,  in  the  year  1844,  by  Charles  Davies,  in  tlie  Clerl  s 
Office  of  tlie  District  Court  of  the  United  States,  in  and  for  the  Soutliern  District  oi 
New  York. 


Engineenng  & 

Mathematical 

Sdences 

Jjbrary 


PREFACE 


The  Treatise  on  Algebra,  by  M.  Bourdon,  is  a  work  of 
singular  excellence  and  merit.  In  France,  it  is  one  of  the 
leading  text  books.  Shortly  after  its  first  pubUcation,  it  passed 
through  several  editions,  and  has  formed  the  basis  of  every 
subsequent  work  on  the  subject  of  Algebra. 

The  original  work  is,  however,  a  full  and  complete  treatise 
on  the  subject  of  Algebra,  the  later  editions  containing  about 
eight  hundred  pages  octavo.  The  time  which  is  given  to 
the  study  of  Algebra,  in  this  country,  even  in  those  semin- 
aries where  the  course  of  mathematics  is  the  fullest,  is  too 
short  to  accomplish  so  voluminous  a  work,  and  hence  it  has 
been  found  necessary  either  to  modify  it,  or  to  abandon  it  al- 
together. 

The  following  work  is  abridged  from  a  translation  of  M. 
Bourdon,  made  by  Lieut.  Ross,  now  the  distinguished  pro- 
fessor of  mathematics  in  Kenyon  College,  Ohio. 

The  Algebra  of  M.  Bourdon,  however,  has  been  regarded 
only  as  a  standard  or  model.  The  order  of  arrangement,  in 
many  parts,  has  been  changed ;  new  rules  and  new  methods 
have  been  introduced :  and  all  the   modifications   which  have 


4  PREFACE. 

been  suggested  by  teaching  and  a  careful  comparison  with 
other  standard  works,  have  been  freely  made.  It  would,  per- 
haps, not  be  just  to  regard  M.  Bourdon  as  responsible  for 
the  work  in  its  present  form. 

It  has  been  the  intention  to  unite  in  this  work,  the  scien- 
tific discussions  of  the  French,  with  the  practical  methods 
of  the  English  school ;  that  theory  and  practice,  science  and 
art,  may  mutually  aid  and  illustrate  each  other. 

CHARLES  DAVIES. 

West  Point,  June,  1844 


CONTENTS. 


CHAPTER  I. 

PRELIMINARY    DEFINITIONS    AND    REMARKS. 

ARTICLBa 

Algebra — Definitions — Explanation  of  the  Algebraic  Signs ^ . .  1 — 28 

Similar  Terms — Reduction  of  Similar  Terms 28 — 30 

Problems — Theorems — Definition  of — Question 31 — 33 

CHAPTER  n. 

OF    ADDITION,    SUBTRACTION,    MULTIPLICATION,    AND    DIVISION. 

Addition— Rule 33—36 

Subtraction — Rule — Remark 36 — 40 

Multijdication — Rule  for  Monomials 40 — 42 

Rule  for  Polynomials  and  Signs 42 — 45 

Remarks — Theorems  Proved 45 — 48 

Of  Factoring  PoljTiomials 48 — 49 

Division  of  Monomials — Rule 49 — 52 

Signification  of  the  Symbol  ao 52 — 54 

Division  of  Polynomials — Rule 54 — 56 

Remarks 57 — 61 

When  m  is  entire,  a"*—b"*  is  divisible  by  a—b 61 — 62 

CHAPTER  HI. 

ALGEBRAIC    FRACTIONS. 

Definition — Entire  Quantity — Mixed  Quantity 62 — 65 

Reduction  of  Fractions 65 — 69 

To  Reduce  a  Fraction  to  its  Simplest  Form 70 

To  Reduce  a  Mixed  Quantity  to  a  Fraction 71 


6  CONTENTS. 

ABTICT.ES 

To  Reduce  a  Fraction  to  an  entire  or  Mixed  Quantity 72 

To  Reduce  Fractions  to  a  Common  Denominator 73 

To  Add  Fractions 74 

To  Subtract  Fractions 75 

To  Multiply  Fractions 76 

To  Divide  Fractions 77 

Ri'sults  from  adding  to  both  Terms  of  a  Fraction 78 

CHAPTER  IV. 

EQUATIONS    OF    THE    FIRST    DEGREE. 

Definition  of  an  Equation — DiiTerent  Kinds — Properties  of  Equations  79 — 86 

Principles  in  the  Solution  of  Equations — Axioms 86 — 87 

Transformation  of  Equations — First  and  Second 87 — 92 

Resolution  of  Equations  of  the  First  De£;ree — Rule 92 — 94 

Questions  involving  Equations  of  the  First  Degree 94 — 95 

Equations  with  two  or  more  Unknown  Quantities 95 — 96 

Elimination — By  Addition — By  Subtraction — By  Comparison 96 — 103 

Indeterminate  Problems — Questions  involving  two  or  more  unknown 

Quantities 103—104 

Theory  of  Negative  Quantities — Explanation  of  the  Terms  Nothmg 

and  Infinity 104 — 114 

Inequalities 114 — 116 

CHAPTER  V. 

EXTRACTION    OF    THE    SQUARE    ROOT    OF     NUMBERS. FORMATION     OF 

THE    SQUARE    AND    EXTRACTION    OF    THE     SQUARE    ROOT  OF    ALGE- 
BRAIC QUANTITIES. 

Extraction  of  the  Square  Root  of  Numbers 116—118 

Of  Incommensurable  Numbers 118 

Extraction  of  the  Square  Root  of  Fractions 118 — 124 

Extraction  of  the  Square  Root  of  Algebraic  Quantities 124 

Of  Monomials  124—127 

Of  Polynomials 127—130 

Calculus  of  Radicals  of  the  Second  Degree 130 — 132 

Addition  and  Subtraction— Of  Radicals 132—133 

Multiplication,  Division,  and  Transformation 133 — 137 

CHAPTER  VI. 

EQUATIONS    OF    THE    SECOND    DEGREE. 

Equations  of  the  Second  Degree 137 — 139 

Involving  two  Terms 139—110 

Complete  Equations  of  the  Second  Degree 14ii — 141 


CONTENTS.  7 

ARTICLES 

Discussion  of  Equations  of  the  Second  Decree 141 — 130 

Problem  of  the  Lights 150—151 

Of  Trinomial  Equations 151 — 154 

Extraction  of  the  Square  Root  of  the  Binomial  a  ±  VT 154 — 157 

l&quations  with  two  or  more  Unknown  Quantities 157 — 159 


CHAPTER  VII. 

OF    PROPORTIONS    AND    PROGRESSIONS. 

How  Quantities  may  be  compared  together 159 

Arithmetical  Proportion  Defined 159 

Geometrical  Proportion  Defined 159 

Arithmetical  Proportion — Sum  of  Extremes 160 

Arithmetical  Progression — Increasing  and  Decreasing 161 

Value  of  Last  Term — How  to  find  it 162 

How  to  find  last  term  in  a  Decreasing  Series 163 

Sum  of  two  Terms  equi-distant  from  Extremes 164 

To  find  Sum  of  all  the  Terms 164 

General  Formulas 165 

To  find  the  first  Term 165 

To  find  the  Common  Difference 165 

To  find  any  Number  of  Means  between  two  Numbers 166 

The  Whole  makes  a  continued  Series 167 

GEOMETRICAL    PROPORTION 

Ratio  Defined 168 

Proportion  Defined 169 

Antecedents  and  Consequents  Defined 170 

Mean  Proportional  Defined ,  171 

Proportion  by  Inversion — or  Inversely 172 

Proportion  by  Alternation 173 

Proportion  by  Composition ...  174 

Proportion  by  Division 175 

Equi-multiples  are  Proportional 176 

Reciprocal  Proportion  Defined 177 

Product  of  Extremes  equal  to  the  Product  of  the  Means 178 

To  make  a  Proportion  from  four  Quantities 179 

Square  of  middle  Term  equal  to  Product  of  Extremes 180 

Four  Proportionals  are  in  Proportion  by  Alternation 181 

Proportion  by  Equality  of  Ratios 182 

Four  Proportionals  are  Proportional  Inversely 183 

Four  Proportionals  are  in  Proportion  by  Composition  or  Division  1^4 

Equi-multiples  have  the  same  Ratio  as  the  Quantities ]S5 

Proportion  by  augmenting  Antecedent  and  Consequent 186 


8  CONTENTS. 

ARTrC!  ES 

Proportion  by  idding  Antecedent  and  Consequent 187 

The  Powers  of  Proportionals  are  in  Proportion 18S 

The  Products  of  Proportionals  are  in  Proportion 189 

Geometrical  Progression  Defined    190 

Value  of  the  Last  Term ■ 191' 

To  find  the  Sum  of  the  Series 192 

To  find  the  Sum  of  the  Terms  of  a  Decreasing  Progression 193 

When  the  Sum  becomes  ^ 194 

To  find   any  Number  of  mean   Proportionals  between   two  Num- 
bers    19f) 

Progressions  having  an  Infinite  Number  of  Terms 196 


CHAPTER  VITI. 

FORMATION  OF  POWERS,  AND  EXTRACTION  OF  ROOTS  OF  ANY  DE- 
GREE WHATEVER. CALCULUS  OF  RADICALS. INDETERMINATE  CO- 
EFFICIENTS. 

Formation  of  Powers   197 — 199 

Theory  of  Permutations  and  Combinations 199 — 202 

Binomial  Theorem 202—206 

Consequences  of  Binomial  Theorem 206 — 209 

Extraction  of  the  Cube  Roots  of  Numbers 209 — 213 

To  Extract  the  nth  Root  of  a  Whole  Number 213—215 

Extraction  of  Roots  by  Approximation 215 — 218 

Cube  Root  of  Decimal  Fractions 218 

Any  Root  of  a  Decimal  Fraction 219 

Formation  of  Powers  and  Extraction  of  Roots  of  Algebraic  Quan- 
tities    220 

Of  Monomials— Of  Polynomials 221—224 

Calculus  of  Radicals — Transformation  of  Radicals 224 — 227 

Addition  and  Subtraction  of  Radicals 227 

Multiplication  and  Division 228 

Formation  of  Powers  and  Extraction  of  Roots 229 

Different  Roots  of  Unity 230—232 

Modifications  of  the  Rules  for  Radicals 232 

Theory  of  Exponents 233 

Multiplication  of  Quantities  with  any  Exponent 234 

Division 235 

Formation  of  Powers 23f 

Extraction  of  Roots 237 

Method  of  Indeterminate  Co-efficients 238—24? 

Recurring  Series , 243 

Binomial  Theorem  for  any  Exponent 244 — 24? 

Applications  of  the  Binomial  Theorem 245— 24t= 


CONTENTS.  9 

ABTICLES 

Summation  of  Series 246 

Summation  of  Infinite  Series 247 


CHAPTER  IX. 

CONTINUED    FRACTIONS. EXPONENTIAL    QUANTITIES. LOGARirHiM-S 

FORMULAS    FOR    INTEREST. 

Continued  Fractions 248 — 254 

Expon-eiilial  Quantities 255 

Theory  of  Logarithms 256—258 

Multiplication  and  Division 258 — 260 

Formation  of  Powers  and  Extraction  of  Roots 260 — 262 

General   Properties 262—266 

Logarithmic  and  Exponential  Series — Modulus 266 — 270 

Transformation  of  Series 270 — 272 

Of  Interpolation 272 

Of  Interest 273 


CHAPTER   X. 

GENERAL  THEORY  OF  EQUATIONS. 

General  Properties  of  Equations 274 — 285 

Of  the  Greatest  Common  Divisor 285—294 

Transformation  of  Equations 294 — 296 

Remarks  on  Transformations 296 

Derived  Polynomials 297 — 300 

Equal  Roots 300—303 

Elimination 303 

By  Means  of  Indeterminate  Multipliers    304 

By  Means  of  the  Common  Divisor 305 — 307 

Method  of  finding  the  Final  Equation 307 — 309 


CHAPTER  XI. 

RESOLUTION    OF    NUMERICAL    EQUATIONS. STURIWs'    THEOREM. 

General   Principles 309 

First  Principle 310 

Second   Principle 311 

Third   Prineii)le 313 

Limits  (if  Heal  Roots 314—317 

Ordinary  Limits  of  Positive  Roots 317 


10  CONTENTS. 

ARTICtES 

Smallest  Limit  in  Entire  Numbers 318 

Superior  Limit  of  Negative  Roots — Inferior  Limit  of  Positive  and 

Negative  Roots 319 

Consequences 320— 32~ 

Descartes' Rule 327—330 

Commensurable  Roots  of  Numerical  Equations 330 — 333 

Sturms'  Theorem 333—34 1 

Young's  Method  of  resolving  Cubic  Equations 342 — 345 

Method  of  Resolving  Higher  Equations 3  J5 


ELEMENTS 


ALGEBRA. 


CHAPTER  I. 

PRELIMINARY    DEFINITIONS    AND    REMARKS. 

1.  Quantity  is  a  general  tenii  applied  to  everything  which 
can  be  estimated  or  measured. 

2.  Mathematics  is  the  science  which  treats  of  the  properties 
and  relations  of  quantities. 

3.  Algebra  is  that  branch  of  mathematics  in  which  the  quanti- 
ties considered  are  represented  by  letters,  and  the  operations  to 
be  performed  upon  them  are  indicated  by  signs.  The  letters  and 
signs  are  called  symbols. 

4.  The  sign  +,  is  called  plus,  and  when  placed  between  two 
quantities  indicates  that  they  are  to  be  added  together.  Thus, 
9  4-  5  is  read,  9  plus  5,  and  indicates  that  the  quantity  repre- 
sented by  5  is  to  be  added  to  the  quantity  represented  by  9. 

In  like  manner,  a  +  ^  is  read,  a  plus  h,  and  denotes  that  the 
quantity  represented  by  b  is  to  be  added  to  the  quantity  repre- 
sented by  a. 

5.  The  sign  — ,  is  called  minus,  and  indicates  that  one  quantity 
is  to  he  subtracted  from  another.  Thus,  9  —  5  is  read,  9  minus  5 
or  9  diminished  by  5. 

In  like  manner,  a  —  5    is  read,  a  minus  b,  or  a  diminished  by  i 

6.  The  sign  X,  is  called  the  sign  of  multiplication.  When 
placed  between  two  quantities,  it  denotes  that  they  are  to  be  mul- 
tiplied together.  Thus,  36  x  25,  denotes  that  36  is  to  be  multi- 
plied by   25.      The  multiplication  of  two  quantities  may   also   be 


12  ELEMENTS    OF    ALGEBRA.  [CHAP.  I. 

indicated  by  placing  a  point  between  them.  Thus,  36.25  is  the 
same  as  36  X  25,  and  is  read,  36  multiplied  by  25,  or  the  prod- 
uct of  36  by  25. 

7.  The  multiplication  of  quantities,  which  are  represented  by 
letters,  is  generally  indicated  by  simply  writing  the  letters  one 
after  the  other,  without  interposing  any  sign.     Thus, 

ah    is  the  same  as    a  X  b,   or  as  a.b ; 
and    ahc,   the  same  as    a  X  b  x  c,    or  as    a.b.c. 
It   is    plain   that  the   notation    ah,  or    ahc,   cannot   be    employed 
when  the  quantities  are  represented  by  figures.     For,  if  it  were  re- 
quired to  express  the  product  of  5  by  6,  we  could  not  write  5  6, 
without  confounding  the  product  with  the  number  56. 

8.  In  the  product  of  several  letters,  as  ahc,  the  single  letters, 
a,  h,  and  c,  are  called  factors.  Thus,  in  the  product  ah,  there 
are  two  factors,  a  and  b ;  in  the  product  acd,  there  are  three, 
a,  c,  and  d. 

9.  There  are  three  signs  used  to  denote  division.     Thus, 

a  -^  b  denotes  that  a  is  to  be  divided  by  b. 

-T-  denotes  that  a  is  to  be  divided  by  b. 

b 

a\h         denotes  that  a  is  to  be  divided  by  h. 

10.  The  sign  =,  is  called  the  sign  of  equality,  and  is  read,  is 
equal  to.  When  placed  between  two  quantities,  it  denotes  that 
they  are  equal  to  each  other.  Thus,  9  —  5  =  4:  that  is,  9  mi- 
nus 5  is  equal  to  4  :  Also,  a  -\-  h  =l  c,  indicates  that  the'  sum  of 
the  quantities  represented  by  a  and  h,  is  equal  to  the  quantity  de- 
noted by  c. 

11.  The  sign  >,  is  called  the  sign  of  inequality,  and  is  used 
to  express  that  one  quantity  is  greater  or  less  than  another. 

Thus,  a  >  i  is  read,  a  greater  than  h ;  and  c  <  J  is  read, 
a  less  than  h ;  that  is,  the  openmg  of  the  sign  is  turned  toward 
the  greater  quantity. 

12.  If  a  quantity  is  added  to  itself  several  times,  as 

a-{-a-i-a-\-a-{-a, 
ii  is  generally  written  but  once,  and  a  figure  is  then  placed  before 
it,  to  show  how  many  times  it  is  taken.     Thus, 
a-\-a-{-a-{-a-\-a^=  5a. 


CHAP.   I.]  DEFINITIONS    AND    REMARKS.  13 

The   number  5  is  called  the  co-efficient  of  a,  and  denotes  that  a  is 
taken  5  times. 

Hence,  a  co-efficietit  is  a  number  prefixed  to  a  quantity,  denoting 
the  number  of  times  which  the  quantity  is  taken.  The  co-efficient 
also  indicates  the  number  of  times  plus  one,  that  the  quantity  is 
added  to  itself.  When  no  co-efficient  is  written,  the  co-efficient 
1  is  alwa)fs  understood.     Thus,    o  =  la. 

13.  If  a  quantity  be  multiplied  continually  by  itself,  as 

axaXaXaXa, 
the  product  is  generally  expressed  by  writing  the  letter  once,  and 
placing  a  number  to  the  right  of,  and  a  little  above  it :    thus, 
a  X  a  X  a  X  a  X  a  =  a^. 

The  number  5  is  called  the  exponent  of  a,  and  denotes  the 
number  of  times  which  a  enters  into  the  product  as  a  factor. 

Hence,  the  exponent  of  a  quantity  shows  how  many  times  the 
quantity  is  a  factor.  It  also  indicates  the  number  of  times,  plus 
one,  that  the  quantity  is  to  be  multiplied  by  itself.  When  no  ex- 
ponent is  written,  the  exponent  1  is  always  understood. 

14.  The  product  resulting  from  the  multiplication  of  a  quantity 
by  itself  any  number  of  times,  is  called  the  power  of  that  quantity : 
and  the  exponent  denotes  the  degree  of  the  power.     For  example, 

a^  =z  a  is  the  first  power  of  a, 
a^  =  a  X  a  is  the  second  power,  or  square  of  a, 
a^  z=  a  X  a  X  a  is  the  third  power,  or  cube  of  a, 
a*  =  aXaXaXa   is  the  fourth  power  of  a, 
a^  =  aXaXaXaXa  is  the  fifth  power  of  a, 
in  which  the  exponents  of  the  powers  are,  1,  2,  3,  4,  and  5;  and 
the  powers  themselves,  are  the  results  of  the   multiplications.     It 
ehoidd  be  observed  that  the  exponent  of  a  power  is  always  greater 
by  unity  tha-n  the  number  of  multiplications. 

15.  As  an  example  of  the  use  of  the  exponent  in  algebra,  lei 
it  be  required  to  express  that  a  number  a  is  to  be  multiplied 
three  times  by  itself,  that  this  product  is  then  to  be  multiplied 
vhree  times  by  b,  and  this  new  product  twice  by  c ;  we  should 
write 

axaxaxaxbxhxbxcxc=z  a*b^c^. 
If  it  were  further  required  to  repeat   this  result  a  certain  num- 


14  ELEMENTS    OF    ALGEBRA.  [CHAP.  I. 

ber  of  times,  say  seven  times,  that  is,  to  add  it  to  itself  six  timtSy 
we  should  simply  Avrite 

This  example  shows  the  brevity  of  the  algebraic  language. 

16.  The  root  of  a  quantity,  is  a  quantity  which  being  multi- 
plied by  itself  a  certain  nmnber  of  times,  will  produce  the  given 
quantity. 

The  sign  y  ,  is  called  the  radical  sign,  and  when  prefixed 
to  a  quantity,  indicates  that  its  root  is  to  be  extracted.     Thus, 

Y  a    or  simply  y  a    denotes  the  square  root  of  a. 

V  a     denotes  the  cube  root  of  a. 
a    denotes  the  fourth  root  of  a. 

The  number  placed  over  the  radical  sign  is  called  the  index 
of  the  root.  Thus,  2  is  the  index  of  the  square  root,  3  of  the 
cube  root,  4  of  the  foiuth  root,  &c. 

17.  The  reciprocal  of  a  quantity,  is  unity  divided  by  that  quan- 
tity.    Thus, 

—  is  the  reciprocal  of  a; 
a 

and,  r  is  the  reciprocal  of  a  +  i. 

a  -\-  b 

18.  Every  quantity  written  in  algebraic  language,  that  is,  with 
the  aid  of  letters  and  signs,  is  called  an  algebraic  quantity,  or  the 
algebraic  expression  of  a  quantity.     Thus, 

is  the  algebraic  expression  of  three  times  the 

quantity  denoted  by  a ; 

.  2  J     is  the   algebraic   expression  of  five  times  the 

(        square   of  a  ; 

_  3,2  ^     is  the  algebraic  expression  of  seven  times  the 

product  of  the  cube  of  a  by  the  square  of  b ; 

is  the  algebraic   expression  of  the  difference 

between  three  times  a  and  five  times  b ; 

is  the  algebraic  expression  of  twice  the  square 

.,  ,        o  1    ,    ^19  1        of  a,  diminished  by  three  times  the    product 
'Za^  —  oab  -\-  Ab^^  r  ,  i  1      ,.         •  1 

of  a  by  0,  augmented  by  fom*  times  the  square 

of  b. 


la  S 


5a  —  55  j 


CriAP.  I.]  DEFINITIONS    AND    REMARKS.  16 

19.  A  single  algebraic  expression,  not  connected  with  any  other 
by  the  sign  of  addition  or  subtraction,  is  called  a  monomial^  or 
simply,  a  term. 

Thus,       3a,    Sa^,    ld?lP-,  are  monomials,  or  single  terms. 

20.  An  algebraic  expression  composed  of  two  or  more  terms 
separated  by  the  sign   +   or   — ,  is  called  a  polynomial. 

For  example,    3a  —  5b,   and  2a2  —  3cb  +  4Z»2,    are  polynomials. 
A  polynomial  composed  of  two  terms,  is  called  a  bino?mal ;  and 
a  polynomial  of  three  terms  is  called  a  trinotnial. 

21.  The  numerical  value  of  an  algebraic  expression,  is  the  num- 
ber obtained  by  giving  a  particular  value  to  each  letter  which  en- 
ters it,  and  performing  the  arithmetical  operations  indicated.  This 
numerical  value  will  depend  on  the  particular  values  attributed  to 
the  letters,  and  will  generally  vary  with  them. 

For  example,  the  numerical  value  of  2a^,  will  be  54  if  we  make 
a  =  3  ;    for,    3^  =  27,    and   2  x  27  =  54. 

The  numerical  value  of  the  same  expression  is  250  when  we 
make  a  =  5;  for,  5^  —  125,  and  2  X  125  =  250. 

22.  It  has  been  said,  that  the  numerical  value  of  an  algebraic 
expression  generally  varies  with  the  values  of  the  letters  which 
enter  it :  it  does  not,  however,  always  do  so.  Thus,  in  the  ex- 
pression a  —  b,  so  long  as  a  and  b  are  increased  or  diminished 
by  the  same  number,  the  value  of  the  expression  will  not  be 
changed.  * 

For  example,  make   a  ==  7   and   i  =  4  :  there  results   a  —  h  =  Z. 
Now,  make    a  =  7  +  5  =  12,    and    />  =  4  +  5  =  9,    and    there 
results,  as  before,    a  —  b  =.  \2  —  9  =  3. 

23.  Of  the  different  terms  which  compose  a  polynomial,  some 
are  preceded  by  the  sign  +,  and  others  by  the  sign  — .  The 
first  are   called  additive  terms,  the  others,  suhtractive  terms. 

When  the  first  term  of  a  polynomial  is  plus,  the  sign  is  gener- 
ally omitted ;  and  when  no  sign  is  written,  it  is  always  under- 
stood to  be  affected  by  the  sign  -)-. 

24.  The  numerical  value  of  a  polynomial  is  not  affected  by- 
changing  the  order  of  its  terms,  provided  the  signs  of  all  the 
terms  remain  unchanged.     For  example,  the  polynomial 

4a3  _  3a26  +  bac"^  =  5ac2  —  'ia'^h  +  4a3  =  —  Za^b  f  bac^  +  4cz* 


16  ELEMENTS    OF    ALGEBRA,  [CHAP.  I. 

25.  Each  of  the  literal  factors  which  compose  a  term,  is  called 
a  dimension  of  the  term ;  and  the  degree  of  a  term  is  the  number 
of  these  factors  or  dimensions.     Thus, 

3a     is  a  term  of  one  dimension,  or  of  the  first  degree. 

bob     is  a  term  of  two  dimensions,  or  of  the  second  degree. 

la^hc^  =  laaahcc   is  of  six  dimensions,  or  of  the  sixth  degree. 

In  general,  the  degree,  or  the  number  of  dimensions  of  a  term,  is 
determined  by  taking  the  sum  of  the  exponents  of  the  letters  which 
enter  this  term.  For  example,  the  term  8a^bcd^  is  of  the  seventh 
degree,  since  the  sum  of  the  exponents,    2+1+1  +  3  =  7. 

26.  A  polynomial  is  said  to  be  homogeneous,  when  all  its  terms 
are  of  the  same  degree.     The  polynomial 

3a  —  2J  +  c  is  of  the  first  degree  and  homogeneous. 

—  4ab  +  b^  is  of  the  second  degree  and  homogeneous. 

50^0  —  4c^  +  2c'^d  is  of  the  third  degree  and  homogeneous. 

8a^  —  4ab  +  c  is  not  homogeneous. 

27.  A  vinculum  or  bar  ,  or  a  parenthesis  (  ),  is  used  to 

express  that  all  the  terms  of  a  polynomial    are   to  be   considered 

together.      Thus,    a  -{-  b  -^  c  X  b,     or     {a  -\-  b  -\-  c)  x  b     denotes 
that  the  trinomial    a  -\-  b  -\-  c    is  to  be  multiplied  by  b ;    also, 

a  +  b-\-  c  X  c  +  d-irf   or    {a  +  b  +  c)  X  {c  +  d  +/) 
denotes  that  the   trinomial    a  -^  b  -^  c    is    to  be   multiplied  by  the 
trinomial    c  -\-  d  -\-  f 

When  the  parenthesis  is  used,  the  sign  of  multiplication  is  usually 
omitted.     Thus,      {a  +  b  -\-  c)  X  b    is  the  same  as  {a  -{-  b  -\-  c)h 
The  bar  is  also  sometimes  placed  vertically.     Thus, 

-\-  a   X  

+  i       is  the  same  as   (a  +  i  +  c)  a?,   or,   a  +  i  +  c  X  a;. 
+  c 

28.  Those  terms  of  a  polynomial  which  are  composed  of  the 
same  letters,  affected  with  the  same  exponents,  are  called  similar 
terms. 

Thus,  in  the  polynomial 

lab  +  Zab  —  AaW  +  Sa'i*, 
the  terms    lab  and   Zab,  are    similar,   and  so   also    are    the    terms 
—  4a'''Z»2   and  ba^b"^,  the    letters  and    exponerts  in   each   being    the 
same.     But  in  the  binomial 

8a26  +  7at2, 


CHAP.   I.J  DEFINITIONS    AND    RKMAUKS.  17 

the.  terms  are  not  similar  ;  for,  although  they  are  composed  of  the 
same  letters,  yet  the  same  letters  are  not  ulTected  with  the  same 
exponents. 

29.  When  a  polynomial  contains  several  similar  terms,  it  may 
often  be  reduced  to  a  simpler  form. 

Take  the  polynomial  Aa^b  —  Sa^c  +  ^a^c  —  2a'^b. 

It  may  be  written  (Art.  24)    Aa^b  —  2a'^b  -f  7a^c  —  Sa'^c. 
But    4(2-6  —  2a'^b   reduces  to  2a-6,    and    7a^c  —  3a^c   to  Aa^c. 
Hence,    4a^  —  3a^c  +  la'^c  —  2a^b  =  2d^b  +  Aa^c. 
When  we  have  a  polynomial  having  similar  terms,  as 
+  2a^bc^  —  Ad^bc^  +  &a%c^  —  8a^c^  +  lla^c^, 
anite  the  additive  and  subtractive  terms  separately :  thus, 

Additive  terms.  Subtractive  terms. 

+    2a3ic2  —    4a3ic2 

+    6a^(y^  —    8a^c^ 

Hence,  the  given  polynomial  reduces  to 

\9a^c^  —  12a3ic2  =  7a^c^. 
It  may  happen  that  the  co-efllcient  of  the  subtractive  term,  ob- 
tained  as    above,  will  exceed  that  of  the  additive  term.     In   that 
case,  subtract   the  positive   co-efficient  from   the   negative,  prefix  tria 
minus  sign   to  the  remainder,  and  then  annex  the  litsral  part. 
In  the  poljTiomial  2d^b  +  2^2/,  _  ^a^h  _  ^a^b 

+  Za%  —  5d^b 

+  2a26  _  3a^ 


+  Hci'^b  —  8aH 

H'lt,  —  8aH  =  —  ba^b  —  2a^b  :    hence 

5a^—  8a^  =  5aH  —  5a^b  —  Sa^i  z=  —  3a^b. 
Hence,  for  the  reduction  of  the  similar   terms  of  a  poljmomial, 
we  have  the  following 

RULE. 

I.  Add  together  the  co-efficients  of  all  the  additive  terms,  and  an- 
nex to  their  sum  the  literal  part:  form  a  single  subtractive  term  tn 
the  same  manner. 

II.  Then,  subtract  the  less  co-efficient  from  the  greater,  and  to  the 
remainder  prefix  the  sign  of  the  greater  co-efficient,  and  aimex  the 
literal  part. 

2 


18  ELEMENTS    OF    ALGKRRA.  [CHAP.   1 

EXAMPLES. 

1.  Reduce  the  pol^Tiomial  \(P-h  —  8«~/;  —  9a^5  -f-  Wcrh  to  its 
simplest   form.  Ans.    —  liP-h. 

2.  Reduce  the  polynomial  lahc^  —  nhc^  —  lahc?  —  Sabc^  +  ^ahc^ 
U)  lis  simplest  form.  Ans.   —  Sabc"^. 

3.  Reduce    the    polynomial      9iP  —  Bac"^  +  ]5cP  -\-  Sea  +  Oi/c^ 

—  24c53    to  its  simplest   form.  Ans.  ac^  -\-  8ca. 

4.  Reduce  the  polynomial  6nc~  —  5aP  -\-  lac''-  —  3aP  —  13ac^ 
4-  \8ab^    to  its  simplest  form.  Ans.   lOah^. 

5.  Reduce     the    polynomial  ahc^  —  abc  +  ^ac^  —  Qabc"^  +  6a^c 

—  8ac^    to  its  simplest  form.  A?is.   —  Sabc"^  -I-  5abc  — Sac^. 

Remark. — It  should  be  observed  that  the  reduction  affects  only 
the  co-efficients,  and  not  the  exponents. 

The  reduction  of  similar  terms  is  an  operation  peculiar  to  al- 
gebra. Such  reductions  are  constantly  made  in  Algebraic  Addition^ 
Subtraction,  Multiplication,  and  Division. 

30.  In  the  operations  of  algebra,  there  are  two  kinds  of  quan- 
tities which  must  be  distinguished  from  each  other,  viz. 

1st,  Those  whose  values  are  known  or  given,  and  which  are 
called  known  quantities ;    and 

2dly,  Those  whose  values  are  unknovro,  which  are  called  un- 
knmon  quantities. 

The  known  quantities  are  represented  by  numbers  and  the  first 
letters  of  the  alphabet,  a.  b,  c,  d,  &c.  ;  and  the  unknown,  by  the  final 
letters,  x,  y,  z,  &c. 

31.  A  problem  is  a  question  proposed  which  requires  a  solution. 
It  is  said  to  be  solved  when  the  values  of  the  quantities  sought 
are  discovered  or  found. 

\  theorem  is  a  gen(>ral  truth,  which  is  proved  by  a  course  of 
reasoning  called  a  demonstration. 

32.  The  following  questi;»n  will  tend  to  show  the  utility  of  the 
algebraic  analysis, 

Quesllon. 

The  siim  of  two  numbers  is  67,  and  their  dijTerfnce  19  ;  what  art 
(hj-  two   numbers? 


CHAP.   I.]  DEFINITIONS    AND    REMARKS.  19 

Solution. 
Let  us  first  establish,  by  the  aid  of  the  algebraic  symbols,  the 
connexion  which   exists   between   the   given   and   unknown    num- 
bers of  the  question. 

If  the  least  of  the  two  numbers  were  knotvn,  the  greater  could 
be  found   by  adding  to  it  the   difference   19 ;    or   in   other  words, 
the  less  number,  plus  19,  is  equal  to  the  greater. 
If,  then,  we  make       x  =  the  less  number, 

X  -f-  19  =r  the  greater, 
and  2x  -{-  19  =  the  sum. 

But  from  the  enunciation,  this  sum  is  to  be  equal  to  67.     There- 
fore we  have 

2a;  -{-  19  =  67. 
Now,  if  2x  augmented  by  19,   gives  67,  2x  alone  is   equal   to 
67  minus  19,  or 

2a;  =  67  —  19,    or  performing  the  subtraction,  2x  =  48. 
Hence,    x  is  equal  to  half  of  48,  that  is, 

X  1=  —  r=  24. 

The  least  number  being  24,  the  greater  is 
x+  19  r=:24  -f  19  =  43. 
And,  indeed,  we  have 

43  +  24  =  67,    and    43  —  24  =  19. 

Another  Solution. 

Let  X  represent  the  greater  number  ; 

then,  X  —  19   will  represent  the  less 

and  2x  —  19  =  67  ;    whence,  2x  =  67  +  19  =  86 ; 

86 
therefore,  x  =:-—=:  43  =  the  greater, 

and  consequently,     x  —  19  =:  43  —  19  =  24  =  the  less 

General  Solution  of  this  Problem. 
The  sum  of  two  numbers  is  a,  and  their  difference  is  b.     Whas 
are  the  two  numbers  ? 

Let  X  =   the  less  number ; 

then  will.  x  -j-  5  =   the  greater. 


20  ELEMENTS    OF    ALGEBRA.  [CHAP.  I. 

Then,  by  the  conditions  of  the  question 

2a:  +  ^  =  a,  the  sum  of  the  numbers  ; 

therefore,     2a;  =  a  —  i   and   x  — = . 

And  by  adding  h  to  each  side  of  the  equality,  we  obtain  the 
greater  number, 

.7         a  ^     .    ,         a  ^ 

^  2         2   ^  2         2 

Hence  we  have 

^    .         a         b  .  , 

x-\-b=.  —  -\—-=i  the  greater  number, 

and  a?  = =  the  less. 

2         2 

As  the  y^rm  of  these  results  is  independent  of  any  particular 
values  attributed  to  the  letters  a  and  b,  the  expressions  are  called 
formulas,  and  may  be  regarded  as  comprehending  the  solution  of 
all  questions  of  the  same  nature,  differing  only  in  the  numerical 
values  of  the  given  quantities.     Hence, 

A  formula  is  the  algebraic  enunciation  of  a  general  rule,  or 
princi[)le. 

The  principles  enunciated  by  the  formulas  above,  are  these  : 

Tlie  greater  of  any  two  numbers  is  equal  to  half  their  suin  in- 
creased by  half  their  difference ;  and  the  less,  is  equal  to  half  their 
sum  diminished  by  half  their  difference. 

To  apply  these  formulas  to  the  case  in  which  the  sum  is  237 
and  difference  99,  we  have 

237       99       237  +  99       336       ,^^ 
the  greater  number   —  — — ^-  —  = =  -—  =  1C8 ; 

237       99       237—99       138 
and  the  less  =  — —  — =  —  =    b9  ; 

and  these  are  the  true  numbers  ;    for, 

168  +  69  =  237   which  is  the  given  sum, 

and  168  —  69  =    99   which  is  the  given  difference. 

From  the  preceding  explanations,  we  see  that  Algebra  is  a 
language  composed  of  a  series  of  symbols,  by  the  aid  of  whicli, 
toe  can  abridge  and  generalize  the  operations  required  in  the  sul'c- 
tion  of  problems,  and  the  reasonings  pursued  in  the  deinonstratwu 
af  theorems 


CHAI      (I.J  ADDITION.  21 


CHAPTER  II. 

OF    ADDITION,    SUBTRACTION,    MULTIPLICATION,    AND    DIVISION. 


ADDITION. 

33.  Addition,  in  algebra,  consists  in  finding  the  simplest  equiv- 
alent expression  for  several  algebraic  quantities.  Such  equivalent 
expression  is  called  their  sum. 

34.  Let   it  be   required  to   add  together  \ 

the  monomials,  j 

(    ,         2c 


The  result  of  the  addition  is     -      -      -      3rt  +  56  +  2c 
an  expression  which  cannot  be  reduced  to  a  more  simple  form. 

Again,  add  together  the  monomials  -       <  20^6' 


The  result,  after  reducing  (Art.  29),  is      -       Vi'fih^ 


Let   it   be  required    to   find  the  sura  of 
rhe  expressions,  ")     2«'  -  3ai  +  P 


3a2  —  4ab 
Sab 
2ab  —  5^2 


Their  sum,  after  reducing  (Art.  29),  is     -     da^  —  5ab  —  4/^^ 

35.  As  a  course  of  reasoning  similar  to  the  above  would  apply 
to  all  algebraic  expressions,  we  deduce,  for  the  addition  of  alge- 
braic quantities,  the  following  general 

RULE. 

I.  Write  down  the  quantities  to  be  added,  with  their  respective 
signs,  so  that  the  similar  terms  shall  fall  under  each  other. 

I I .  Reduce  the  similar  terms,  and  annex  to  the  results  those  terms 
which  cannot  be  reduced,  giving  to  each  term  its  respective  si<rn. 


3«?  —  4a4  —  2^3 
5s2  +  2e4  —   ^ 

8tf2  Jf.    ah  —  5P  —  3c» 


23  ELEMENTS    OF    ALGEBRA  [CHAP.  IL 

EXAMPLES. 

1.  Add  together  the  polynomials, 
3a2  _  262  _  4ab,    5a^  —  P  +  2ab,    and    3ab  —  3c^  —  2b\ 

The  term  Sa^  being  similar  to  Sc^, 
we  write   Sa^   for  the  result  of  the   re- 
liuction  of  these  two  terms,  at  the  same  < 
time  slightly  crossing  them    as  in  the 
terms  of  the  example. 

Passing  then   to  the  term    —  4ab,  which  is   similar  to  the  two 
terms    +  2ab    and    +  3ab,    the   three    reduce    to    +  ab,   which   is 
placed  after  Sa^,  and  the  terms  crossed  like  the  first  term.     Pas 
sing  then  to  the  terms  involving  b^,  we  find  their  sum  to  be  —  5b\ 
after  which  we  ^vrite    —  Sc^. 

The   marks   are   drawn   across   the   terms,    that    none   of  them 
may    be  overlooked  and  omitted. 

(2).  (3). 

7x  +  3ab  +    2c  16a'^b^  -{-    be  —  2abc 

—  3x  —  3ab  —5c  —  4a'^b^  —  96c  -f  6abc 
5x  —  9ab  —9c  —  OaH^  -\-    be  -}-     abc 

Sum.       9a:  —  9ab  —  12c  3aH'^  —  Ibc  +  babe 

(4).  (5). 

a  -\-    ab  —     cd  -{-    f  Gab  +       cd  -\-  d 

3a  4"  5ai  —  &cd  —     f  3ab  -\-     5cd  +  y 

—  5a  —  Sab  -\-  6cd  —  7f  —  4ab  +     6cd  -j-  x 

—  a -{-    ab  -\-     cd -^  4/  —  5ab  —  \2ed  —  y 

—  2a-\-ab  +  0      —  3/  0  0      ^  x  +  d 

6.  Add   together    3a -\- b,     3a  -\- 3b,    —  9a  —  7b,   6a +  96,     and 
S/i  +  36  +  8c.  Ans.  11a  +  96  +  8c. 

7.  Add  together    3aa:  +  3ac  -\-  f,    —  9aa;  -\-  7a  -\-  d,    +  Sax  -}-  3oc 
-h  3/,    8aa;  +  13ac  +  9/,    and    —  14/+  3ax. 

Ans.  llax-{-  19ae -f-i- 7a -\- d. 

8.  Add  together  the  polynomials,    3a'^e  +  5a6,    7a'^c  —  3a6  +  3oc, 
5a^c  —  6a6  +  9ac,   and    —  Sa^c  +  a6  —  12ac.      Ans.  7(P-c  —  3ah. 

9.  Add     the    pol}Tiomials      190^x^6  —  12a3c6,      bd^x^b  +  \5c?ch 
-  lOax,    —  2(r-x%  —  Uahb,    and    —  18a2a;36  —  12a3c6  +  9aa;. 

Ans.  4a'^x^b  —  22a^cb  —  ax 


CHAP.   II.]  ADDiriOV.  23 

10.  Add  together  3a  +  6  +  c,  5a  -f  2i  +  Sac,    a  +  c -{-  ac,   and 

—  3a  —  9ac  —  8b.  Ans.  6a  —  5h  +  2c  —  5ac. 

11.  Add  together    da^b  -\-  6cx  +  9bc",   7 ex  —  8a^b,   and    —  15car 
~  9ic2  4-  20^.  Arts.   —  a^b  —  2cx. 

13.  Add  together    Sax  +  bab  +  Za'^h'^c^,    -  18aa:  +  Ga^  +  lOaA, 
and    lOax  —  Ibab  —  &a^¥c^.  Ans.    —  SaWc^  +  6a2. 

13.  What  is  the  sum  of    AXd^b'^c  —  21abc. —WcP-y    and    lOa^Z/V 
f  Qa^/c?  yln^.  51a^62c  —  18aic  —  Ha^y. 

14.  What  is    the    sura  of    18a6c  —  9ab  +  Gc^  —  3c  +  9aar     and 
9a/jc  +  3c  —  9ax  ?  >!«:?.  27aic  —  9ab  +  Gc^, 

15.  What   is    the    sum   of     8ahc  -\-  Pa  —  2ca;  —  6x1/     and     7cx 

—  xy  —  \3Pa1  Ans.  8abc  —  12b^a  +  5cx  —7xy. 

16.  What    is    the    sum   of    9a"c  —  14a5y  +  ISa^fe^     ^nd     —  d'-c 

—  8aW\  A«.>\  8a2c  —  14aiy  +  laW. 

17.  What  is  the  sum  of    17a^Z»2  ^  9^-/,  _  3^2^    _  14^5^2  _|_  7^^ 

—  9a3,    —  15a36  +  7a^62  —  c^,    and    14a"6  -  \9d^b  ? 

Ans.  . 

18.  What  is  the  sum  of   3aa;^  —  9ax^  —  \laxy,    —  9ax^  +  18ax^ 
•\-  34axy,    and    la^'b  +  Sax^  —  Jax"^  +  46ca;  ?    Ans.  . 

19.  Add  together    3a2  +  ^aWc"^  —  9a^r,    la^  —  8a'^b\^  —  lOa^a., 
10a6  +  IGa^^V  +  19a^x.  Ans.   10a-  +  ISaH^c^  +  lOab. 

20.  Add  together    7«-/y  —  3uic  —  8b^c  —  9c^  +  cd^,    8abc  —  5aH 
-j_  3c3  _  4^2c  _f_  c(l~,    and   4a2J  _  Sc^  +  952c  —  3d^. 

Ans.  6d^b  +  5a5c  —  352c  —  \\c^  -\-  2cd?  —  3d^. 

21.  Add  together    —  18a^  +  2ab^  +  60252  _  8a5*  +  7a^  —  ba'^b^ 

—  5a^b  4-  Gab*  +  lla252,  ^„^,    _  15^35  ^  12a252. 

22.  What    is   the    sum    of     3a352c  —  16a*a;  —  9ax^d,      +  6a^^e 

—  Gax^d  +  17a*a:,    +  16ax^d  —  a^x  —  8a^^^c  ? 

Ans.  aH'^c  +  ax-^d. 

23.  WTiat  is  the  sum  of  the  following  terms :  viz.,    8a^  —  10a*5 
-  16a352  +  4^253  _  i2a4j  ^  15a352  _}_  24a253  —  6ab*  —   16a^^ 

4-  20a253  +  32a5*  —  85^  ? 

Ans.  8a5  —  22a*5  —  17^352  _{_  48a2^,3  _|_  26a5*  —  85* 


24  KLEMENTS    OF    ALGEBRA.  [CHAP.   J3. 

SUBTRACTION. 

36.  Subtraction,  in  Algebra,  consists  in  finding  the  simplest 
expression  for  the  difference  between  two  algebraic   quantities. 

Let  it  be  required  to  subtract  46  from  5a.  Here,  as  the  quan- 
tities are  not  similar,  their  difference  can  only  be  indicated,  and 
we  write 

5a  —  4h. 

Again,  let   it   be    required  to   subtract  4a^5  from    7a^.      These 
('•rms    being  similar,   one   of  them  may  be  taken    from  the  other, 
and  their  true  difference  is  expressed  !)y 
7a^  —  4>Pb  =  3a^. 
For  another  example,  from        -         -         -         4a 

take  the  binomial  -         -         -         -         26  —  3c. 


4a  —  26  +  3c 

True  remainder. 


The  subtraction  may  be  indicated  thus,     -         4a  —  (26  —  3c)  , 
that    is,   the    quantity   to   be    subtracted    may  be 
placed  within  a  parenthesis,  and  written  after  the 
other  quantity,  with  a  minus  sign. 

Now,  in  order  to  express  this  difference  by  a  single  polyno 
raial,   let  us   see  what  the   nature  of  the  question  requires. 

From  4a,  we  are  to  subtract  the  difference  between  26  and  3e, 
and  if  6  and  c  were  given  numerically,  that  difference  would  be 
known  ;  but  since  3c  cannot  be  taken  from  26,  26  is  first  subtracted 
from  4a,  which  gives  4a  —  26.  Now,  in  subtracting  the  number 
of  units  contained  in  26,  the  number  taken  away  from  4a,  is  too 
great  by  the  number  of  units  contained  in  3c,  and  the  result 
4a  —  26  is  therefore  too  small  by  3c;  this  remainder  must  there- 
fore be  corrected  by  adding  3c  to  it.  Hence,  there  results  from 
the   proposed  subtraction    4a  —  26  +  3c. 

37.  Hence,  for  the  subtraction  of  algebraic  quantities,  we  have 
the  following  general 

RULE. 

I.  Write  the  quantity  to  he  subtracted  vnder  that  from  which  it 
ts  to  he  taken,  placing  the  similar  terms,  if  there  are  any,  undn 
each  other. 

II.  Chiinrre  the  sirrns  of  all  the  terms  of  the  pohpiotinnl  to  6c 
subtracted,  or  conceive  them  to  he  chingrd,  and  then  reduce  the  poly 
HOTnial  result  to  its  simplest  form 


CHAP.    II.] 

SUBTRACTION. 
EXAMPLES. 

25 

0)- 

5  2-0 

(1)- 

From 

Sac  —  5ab  -\-  c^ 

s"=l 

6ac  —  5ab  -\-  c^ 

Take 

3ac  +  Sab  —  7c 

2  =  » 
■  M  5 

c-  IS  is 

5   3 

—  Sac  —  Sab  +  7c 

Remainder 

Sac  —  8ab  +  c2  +  7c. 

Sac  —  Sab  +  c2  +  7c 

(2). 

(3). 

From 

16a2  _    56c  +  lac 

19abc  —  IGax  —    5axy 

Take 

14«2  +    5bc  +  Sac 

17  abc  +     7  ax  —  Ibaxy 

Remainder 

2a2  _  106c  —    ac 

2abc  —  2Sax  +  lOaxr/ 

(4). 

(5). 

From 

5a3  _  4aH  +     Sb^c 

4ab  —    cd  -|-  Sa^ 

Take 

-  2a3  +  Sa^  —    SPc 

5ab  —  4cd  +  3a2  +  5b^ 

Remainder 

7a3  —  la'^b  +  ll/>2c 

-    ah  +  Scd  +  0     -  562. 

6.  From    3a2x  —  13a6c  +  7a2,    take    9a^x  —  ISabc. 

Ans.   —  da'^x  -\-  7cP'. 

7.  From      fiXa^b'^c  —  \Sabc  —   Wa'^y,     take      Ma^lfic  —  27ahc 

—  14a2y.  Ans.    lOaWc  +  9abc. 

8.  From    27a6c  —  9ab  -f  6c2,    take    9a6c  +  3c  —  9<7,r. 

Ans.   ISabc  —  9ab  +  6c2  —  3c  +  9ax. 

9.  From    8a5c  —  1253a  +  5cx  —  7xy,    take    7cx  —  xy  —  ISb'^a, 

Ans.  Sabc  -j-  b^a  —  2ca;  —  6xy. 

10.  From     Sa^c  —  14a6y  +  7a262,    take    9a2c  —  I4aby  +  15a262. 

Ans.   —  a'^c  —  8a262. 

11.  From    9a«x2  —  13  +  20a63a;  —  4b^cx'^,    take     Sb^cx^  +  QaV 

—  G  +  Sab^x.  Ans.   17  ab^x  —  7bhx^  —  7. 

12.  From     5a*  —  7aW  —  Sc^d!^  +  7d,    take     3a*  —  3a2  —  7c^J?- 

—  15a362.  Ans.  2a*  +  Sa'^b'^  +  4c^d'^  +  7J  +  3a2. 

13.  From    51a262  _  48a36  +  10a*,    take    10a*  —  Sa^*  —  6a2i2. 

Ans.  b7aW  —  4Qa^b. 

14.  From     21x3y2   _|_    25a;2y3   +    GSxy*   —    40y5,     take     64^2^3 
+  48a-y*  —  40y5.  Ans.  2idxy*  —  S9xhf  +  21a;3y2. 

L5.    From      bSx^y"^  —  Ibx^y"^  —  ISxhj  —  bSx^ ,     take      -   l.^.r'y^ 
r  18a:3y2  _}.  24x*y.  Ans.  35x3y2  _  \2x^y  —  SCx"* 


26  ELEMENTS    OF    ALGEHRA.  [CH^^P.   II. 

38.  From  what  has  preceded,  we  see  that  polynomials  may  l)e 
subjected  to  certain  transformations. 

For  example  -  -  60^  —  3ab  +  2b'^  —  2hc, 

may   be   written  -  6a^  —  [Sab  —  2b'^  +  2bc). 

In  like  manner    -  -  7a^  —  Sa%  —  ib'^c  +  6b\ 

may  be   written  -  7a^  —  {Sa?b  +  Ab^-c  —  W^) ; 

or,  again,     -       -  -  la^  —  Sa^b  —  [Ab'^c  —  6b^). 

Also,     -       -       -  .  8a2  —  6a^^  +  5^253, 

becomes       -       -  -  Sa^  —  [Ga^b^  —  da^b^). 

Also,     -       -       -  -  9a2c3  —  8a*  +  ^»2  _  c, 

may  be  written  -  90^6^  —  (8a*  —  b'^  -\-  c) ; 

or,  it  may  be  written  Qa^c^  +  b"^  —  (8a*  +  c). 

These  transformations  consist  in  decomposing  a  polynomial  into 
two  parts,  separated  from  each  other  by  the  sign   — . 

It  will  be  observed  that  the  sign  of  each  term  which  is  placed 
within  the  parenthesis  is  changed.  Hence,  if  we  have  one  or 
more  terms  included  within  a  parenthesis  having  the  minus  sign 
before  it,  if  the  parenthesis  is  omitted,  the  signs  of  all  the  terms 
must  be  changed. 

Thus,  4a  —  {Gab  —  3c  -^  2b), 

is  equal  to  4a  —    Gab  +  3c  +  2b. 

xVlso  Gab  —  ( —  4ac  -\-  3d  —  4ab), 

is  equal  to  Gab     -|-      4ac  —  3d  -{-  4ab. 

39.  Remark.  —  From  what  has  been  shown  in  addition  and 
subtraction,  we  deduce  the  following  principles. 

1st.  In  Algebra,  the  words  add  and  sum  do  not  always,  as  in 
arithmetic,  convey  the  idea  of  augmentation.  For,  a  —  b,  which 
may  result  from  the  addition  of  —  J  to  «,  is  properly  speaking, 
the  arithmetical  difference  between  the  number  of  units  expressed 
by  a,  and  the  number  of  units  expressed  by  b.  Consequently, 
this  result  is  numerically  less  than  a. 

To  distinguish  this  sum  from  an  arithmetical  sum,  it  is  called 
the  algebraic  sum. 

Thus,  the  polynomial,     2a^  —  3a'^b  +  3b'^c. 
is   an   algebraic   sum,   so   long   as   it  is   considered    as    the   resuU 
of  the  union  of  the  monomials    2^3,    —  Sa^^,    +  3b-c,    with  then 
respective  signs ;  but,  in  its  proper  acceptation,  it  is    the  arithmeli- 


CHAP.   II.]  MULTIPLICATION.  27 

cal  difference  between  the  sum  of  the  units  contained  in  the  ad- 
ditive terms,  and  the  sum  of  the  units  contained  in  the  subtractive 
terms. 

It  follows  from  this,  that  an  algebraic  sum  may,  in  the  numeri- 
cal applications,  be  reduced  to  a  negative  number,  or  a  number 
affected  with  the  sign  — . 

2d.  The  words  subtraction  and  dijference,  do  not  always  convey 
the  idea  of  diminution.  For,  the  difference  between  +  a  and  —  b 
being  a  —  [—  b)  zzl  a  -\-  b,  is  numerically  greater  than  a.  This 
result  is  an  algebraic  difference. 


MULTIPLICATION. 

40.  Algebraic  raultiplication  has  the  same  object  as  arithmeti- 
cal, viz.,  to  repeat  the  multiplicand  as  many  times  as  there  are 
units  in  the  multiplier.  The  multiplicand  and  multiplier  are  called 
factors. 

It  is  proved  in  Arithmetic  (see  Davies'  Arithmetic,  §  26),  that 
the  value  of  a  product  is  not  affected  by  changing  the  order  of  its 
factors  :  that  is, 

I2ab  =  ab  X  12  =  ba  X  12  =  a  X  12  X  b. 
For  convenience,  however,  the  letters  in  each  term  are  generally 
arranged  in  alphabetical  order,  from  the  left  to  the  right. 

Let  it  be  required  to  multiply    7a^b^    by    4a'^b. 

By  decomposing  the  multiplicand  and  multiplier  into  their  fac- 
tors, we  may  write  the  product  under  the  form 

7a^b^  X  4a'^b  =  laaabh  X  Aaah ; 
and  since  we  may  change  the  order  of  the  factors  without  affect- 
ing the  value  of  the  product,  we  have, 

laW  X  4a^  =  7  X  Aaaaaabbb  =  2Sa^P ; 
a   result   which   is   obtained   by  multiplying   the    co-efficients    to- 
gether for   a   new   co-efficient,   and  adding  the   exponents  of  the 
same  letter,  for  the  new  exponents. 

Again:  multiply  the  monomial     12a%^c'^   by    Qa%'^cP. 

We  can  place  the  product  under  the  form, 

■\2a^¥c'^  X  ^aWd^  =  12  X  Saaaaabbbbbbccdd  =  OGa^^c'^d^. 

By  considering  the  manner  in  which  these  results  are  obtained, 
we  see    that   any   quantity,   as   a,   must   be   found   as   many  times 


28  ELEMENTS    OF    ALGEBRA.  [CHAP.  11. 

a  factor  in  the  product,  as  it  is  a  factor  in  both  the  multiplicand 
and  multiplier;  which  number  will  always  be  expressed  by  the 
sum  of  its  exponents. 

41.  Hence,  for  the  multiplication  of  monomials  we  have  the 
following 

RULE. 

I.  Multiply  the  co-efficients  together  for  a  new  co-efficient. 

II.  Write  after  this  co-efficient  all  the  letters  which  enter  into  the 
multiplicand  and  multiplier,  affecting  each  with  an  exponent  equal 
to  the  sum  of  its  exponents  in  both  factors.     • 

EXAMPLES. 

(1)  -       -  8a2ic2     X  labd"^  =  56a^^c^d^. 

(2)  -       -       2\a%Hc  X  8a6r3  _  \QQa^b^cH. 

(3)  (4)  (5)  (6) 

Multiply  -       -         Sa^ft    -     -     I'ia'^x     -     -     6xyz      -     -       a^xy 
by         -       -         25a2    -     -     I2x'^i/     -     -       ay^z     -     -     Ix.y^ 


Ga'^b^  I44a'^x^y  6ax7/^z'^  2a'^x^y^. 

7.  Multiply    Sa^b^c   by   7a%^cd.  Ans.  56a^Wc'^d. 

8.  Multiply   5abd^   by    12cd^.  Ans.  60abcd^. 

9.  Multiply   la^bd'^c'^   by   abdc.  Ans.  la^Pd^c*. 

42.  We  will  now  proceed  to  the  multiplication  of  polynomials. 
In  order  to  explain  the  most  general  case,  we  will  suppose  the 
multiplicand  and  multiplier  each  to  contain  additive  and  subtrac- 
tive  terms. 

Let  a  represent  the  sum  of  all  the  additive  terms  of  the  multi- 
plicand, and  b  the  sum  of  the  subtractive  terms ;  c  the  sum  of 
the  additive  terms  of  the  multiplier,  and  d  the  sum  of  the  sub- 
tractive  terms.  The  multiplicand  will  then  be  represented  by 
a  —  b    and  the  multiplier,  by    c  —  d. 

We  will  now  show  how  the  multiplication  expressed  by 
{a  —  b)  X  (c  —  d)    can  be  eflected. 

The  required  product  is  equal  to  a  —  6 
taken  as  many  times  as  there  are  units 
in  c  —  d.  Let  us  first  multiply  by  c ; 
that  is,  take  a  —  b  as  many  times  as 
there  are  units  in  c.  We  begin  by  wri- 
ting  ac,   which    is   too   great  by   b   taken 


a 

-b 

c 

-d 

ac 

-be 

—  ad  -\-  bd 

ac 

-bc- 

-ad  +  bd. 

CHAP.  II.]  MULTIPLICATION.  29 

c  times  ;  for  it  is  only  the  difference  between  a  and  b,  that  is  first 
to  be  multiplied  by  c.  Hence,  ac  —  be  is  the  product  of  a  —  b 
by  c.  But  the  true  product  is  a  —  b  taken  c  —  d  times :  hence, 
the  last  product  is  too  great  by  a  —  b  taken  d  times ;  that  is, 
by  ad  —  bd,  which  must  be  subtracted.  Changing  the  signs  and 
subtracting  this  from  the  first  product  (Art.  37),  we  have 
(a  —  b)  X  (c  —  d)  =  ac  —  be  —  ad  -{-  bd  : 
If  we  suppose  a  and  c  each  equal  to  0,  the  product  will  re- 
duce to    +  bd. 

43.  By  considering  the  product  of  a  —  h  by  c  —  d,  we  may 
deduce  the  following  rule  for  the  signs,  in  the  multiplication  of 
two  polynomials. 

When  two  terms  of  the  multiplicand  and  multiplier  are  affected 
with  the  same  sign,  their  product  will  be  affected  with  the  sign  +, 
and  when  they  are  affected  with  contrary  signs,  their  product  will  be 
affected  with  the  sign    — . 

Again,  we  say  in  algebraic  language,  that  +  multiplied  by  -\-, 
or  —  multiplied  by  — ,  gives  -\-  ;  —  multiplied  by  +>  or  +  mul- 
tiplied by  — ,  gives  — .  But  since  mere  signs  cannot  be  multi- 
plied together,  this  last  enunciation  does  not,  in  itself,  express  a 
distinct  idea,  and  should  only  be  considered  as  an  abbreviation 
of  the  preceding. 

This  is  not  the  only  case  in  which  algebraists,  for  the  sake  of 
brevity,  employ  expressions  in  a  technical  sense  in  order  to  se- 
cure the  advantage  of  fixing  the  rules  in  the  memory. 

44.  Hence,  for  the  multiplication  of  polynomials  we  have  the 
foUowidg 

RULE. 

Multiply  all  the  terms  of  the  multiplicand  by  each  term  of  the 
tnultiplter  in  succession,  affecting  the  product  of  any  two  terms  with 
the  sign  plus,  when  their  signs  are  alike,  and  with  the  sign  minusy 
when  their  signs  are  unlike.  Then  reduce  the  polynomial  result  to 
its  simplest  form. 

1.  Mwhiply 3a2  +  4aJ  +  £2 

by 2a    -\-  5b 


Ca3  +     8aV}  +     2ab^ 
The  product  after  reducing,  +  \5aH  +  20«Z»2  +  5P 

becomes 6m-<  -f  23u^  +  22irb'^  +  5b^. 


80  ELEMENTS    OF    ALGEBRa.  [CHAP.  II 

(2).  (3). 

a'  +  y^  x^  -r  xy^  +  ^aa- 

ar   —  y  ax  -\-  5ax 


a^  +  xy"^  ax^  4-  ax'^y^  -f-  Va^a;^ 

—  x'^y  —  y^  +  5ax^    +  ^ax'^y^  +  SSa^x' 

g^  +  a:?/^  —  x^y—'^  6ax^^+_6ax^^  j-  42a'^x^. 

4.  Multiply    x"^  +  2(7X  +  a^    by    x  -{-  a. 

Ans.  x^  +  So-r^  +  3a^x  +  a^. 

5.  Multiply    x^  -{-  y^    by     a:  +  y- 

-4^5^.  x^  -f-  xy^  4-  a"2y  -\-  y'. 

G.  Multiply    3ai2  _^  e^^c^    by    3«62  _|_  Sa^c^. 

Ans.  9a^b*  +  27aHh^  +  18a*c*. 

7.  Multiply    4a;2  —  2y    by    2y.  Ans.  8x^y  —  4y2. 

8.  Multiply    2x   +  iy    by    2t  —  4y.  Ans.  4x^'  —  16y2. 

9.  Multiply      a;^  +  x^y  +  xy^  +  y^    by    x  —  y.        Ans.  -^ 

10.  Multiply    x"^  -\-  xy  -\-  y^    by    x"^  —  xy  -{■  y^. 

Ans.  X*  +  x'^y^  +  y*. 
In  order  to  bring  together  the  similar  terms,  in   the  product  of 
two  polynomials,  we  arrange  the  terms   of  each   polynomial  with 
reference  to  a  particular  letter. 

11.  Multiply     4a3  —    5a'^b  —    8ah^ -{- 2b^ 
by  2a2  —     Sab    —    4b^ 

8a5  —  lOa^b  —  IGa^b^  +     4a^b^ 

—  12a*b  +  l5aH^  +  24a2i3  _    Qab* 

—  \(jaW  +  2002^3  +  32ai4  —  86' 
8a^  —  22a*b  —  17 a'b^  +  48^2^,3  ^  26«&^  —  865. 

After  ha-\ang  arranged  the  polynomials,  with  reference  to  the 
letter  a,  multiply  each  term  of  the  first,  by  the  term  20^  of  the 
second;  this  gives  the  polynomial  8a^  —  lOa'^b  ~  Ida^"^ -{- 4a'^b^, 
the  signs  of  which  are  the  same  as  those  of  the  multiplicand. 
Passing  then  to  the  term  —  Sab  of  the  multiplier,  multiply  each 
term  of  the  multiplicand  by  it,  and  as  it  is  affected  with  the 
gign  — ,  affect  each  product  with  a  sign  contrary  to  that  of  the 
corresponding   term  in  the  multiplicand;   this  gives 

—  12a^i  +  I5a^^  +  24a^b^  -  Gab* 


^ 


CHAP.  II.]  MULTIPLICATION.  31 

The  same  operation  is  also  performed  with  the  term  —  4/>»', 
which  is  also  subtractive ;   this  gives, 

—  1 6^352  _j_  20^2^3  _|_  32aJ4  _  8J5, 

The  product  is  then  reduced,  and  we   finally  obtain,  for  the  most 
simple  expression  of  the  product, 

8a^  -  22a*i  —  l7aW  +  48a^^  +  26ab*  —  8b^. 

12.  Multiply    2a2  —  3ax  +  Ax^    by    5a2  _  Qax  -  2x2. 

Ans.   10a*  —  21a?x  +  ZAa^x"^  —  ISax^  —  Bx". 

13.  Multiply    3x2  _  2yx  +  5    by   x'^-\-2xy  —  3. 

Ans.  3x*  +  4x3y  —  4x2  —  4x2y2  +  16x3/  -  15. 

14.  Multiply    3x3  _|.  2x2y2  -f  3^2   by   2x3  _  3^2y2  _j_  5^3. 

Ans  \  ^^^  ~  ^^^y^  ~  ^'^y*  +  ^^^y"  "^  ^  ^^^'^ 

15.  Multiply    Sax  —  Qab  —  c   by    2i7x  +  a^  +  c. 

.4«5.   16a2x2  —  4a26x  —  6a2^2  _|_  Qacx  —  labc  —  c^. 

16.  Multiply    3a2  _  5*2  +  3c2    by    a^  —  ^3. 

Ans.  3a*  —  602^2  4.  ^a'^c'^  —  3^263  +  5J5  _  ^b^c^- 

17.  Multiply   3a2_5irf+    c/ 
by  —  5a2  +  4bd  —  8cf 

Prod.  red.       —  15a*  +  37a2,^J  —  29a\f^^0b^(P  +  AAbcdf  —  8c^^. 

18.  Multiply   4a352  —  5aH^c  +  8a-bc^  —  3a2c3  _  7abc^ 
by  2aJ2    _  4^^^     _  2/^c2     +  c^. 

f         8a*b*     —  lOa^'c  +  2Sa^'^c^  —  34a^b'^c^ 
Prod.  red.  <  —   4a2i3c3  _  IGa*^^^  _^  Ua^c*    +  7a2i2c4 
C  +  ]4a25c5    -}-  I4a^2c5  _    3a2c6      —7abc^. 

45.  Resvlts  deduced  from  the  multiplication  of  polynomials. 

1st.  If  the  polynomials  which  are  multiplied  together  are  ho- 
mogeneous, 

Their  product  will  also  be  homogeneous,  and  the  degree  of  each 
term  will  be  equal  to  the  sum  of  the  degrees  of  any  two  terrnn 
of  the  mvltiplicand  and  multiplier. 

Thus,  in  example  18th,  each  term  of  the  multiplicand  is  of 
the  5th  degree,  and  each  term  of  the  multiplier  of  the  3d  de- 
gree :  hence,  each  term  of  the  product  is  of  the  8lh  degree. 
This  remark  serves  to  discover  any  errors  in  the  addition  of  the 
exponents.* 


82  ELEMENTS    OF    ALGEBRA.  [CHAP.  II 

2(1.  If  no  two  of  tlie  partial  products  are  similar,  there  will 
be  no  reduction   among  the  terms   of  the    entire    product:    hence. 

The  total  number  of  terms  in  the  entire  product  will  be  equal  to 
the  number  of  terms  m  the  multiplicand  multiplied  by  the  number 
of  terms  in  the  multiplier. 

•This  is  evident,  since  each  term  of  the  multiplier  will  produce 
as  many  terms  as  there  are  terms  in  the  multiplicand.  Thus,  in 
example  16th,  there  are  three  terms  in  the  multiplicand  and  two 
in  the  multiplier :  hence,  the  number  of  terms  in  the  product  is 
equal  to    3  X  2  =r  6. 

3d.  Among  the  difTerent  terms  of  the  product,  there  are  always 
some  which  cannot  be  reduced  with  any  others.  For,  let  us 
consider  the  product  with  reference  to  any  letter  common  to  the 
multiplicand  and  multiplier.     Then,  the  irreducible   terms  are, 

1st.  The  term  produced  by  the  multiplication  of  the  two  terms 
of  the  multiplicand  and  multiplier  which  contain  the  highest  ex- 
ponent of  this  letter ;  and  the  term  produced  by  the  multiplica- 
tion of  the  two  terms  which  contain  the  lowest  exponent  of  this 
letter.  For,  these  two  partial  products  will  contain  this  letter, 
affected  with  a  higher  and  lower  exponent  than  either  of  the 
other  partial  products,  and  consequently,  they  cannot  be  similar  to 
any  of  them.  This  remark,  the  truth  of  which  is  deduced  from 
the  law  of  the  exponents,  will  be  very  useful  in  division. 

Multiply       -       -       5«^^>2  +  3a^b  —  ab*  —  2a&3 
by     -       -       -         a'^b    —    ab"^ 

C  5a%^  +  3a^^  —    a^b^  —  2a^^- 

Product,  J  -  5a^b*  -  3a^^  +     a~b^  +  2«^^>^ 

If  we  examine  the  multiplicand  and  multiplier,  with  reference 
to  a,  we  see  that  the  product  of  5a'^b'^  by  a'^b,  must  be  irre- 
ducible ;  also,  the  product  of  —  2ab^  by  ab^.  If  we  consider  the 
letter  b,  we  see  that  the  product  of  —  ab*  by  —  ab"^,  must  be 
irreducible,  also  that  of  Ba'^b   by   a^b. 

46.  We  will  apply  the  rules  for  the  multiplication  of  algebraic 
quantities  in  the  demonstration  of  the  following  theorems. 

THEOREM    I. 

The  square  of  the  sum  of  two  quantities  is  equal  to  the  square 
of  the  first,  plus  twice  the  product  of  the  first  by  the  second,  plus 
the  square  of  the   second.  * 


CHAP,   II.]  MULTIPLICATION'.  33 

Let  a  denote  one   of  the  quantities  and  h  the  other :    then 

a  +  ^''  =    their  sum. 
Now,  Ave   have  from  known  principles, 

[a  +  hf  =  (a  +  J)  X  (a  +  i)  =  c2  +  lah  +  V', 
which  resuh   is   the    enunciation  of  the  theorem  in  the  langiiag 
of  Algebra. 

To  apply  this  result  to  finding  the  square  of  the   binomial 

we  have         (5a2    -f  ^d^hf  =  25a*  +  80a*d  +  64a*^»2. 
Also,  (Go^i  +  9aP)    =  36^8^,2  ^  lOSa^M  +  Sla^js  . 

also,  (8a3    -f7ac6)2=. 

THEOREM    II. 

The  square  of  the  difference  between  two  quantities  is  equal  to 
the  square  of  the  first,  minus  twice  the  product  of  the  first  hy  the 
second,  plus  the   square   of  the   second. 

Let  a  represent  one  of  the  quantities  and  b  the  other :    then 

a  —  5  =    their  difference. 
Now,  we  have  from  known  principles, 

[a  —  by  =  {a  —  b)x{a  —  h)  =  d^—  2ab  +  ^', 
which  is  the  algebraic  enunciation  of  the  theorem. 
To  apply  this  to  an  example,  we  have 

(7^2^,2  _  i2ab^)    =  49a*b*  —  l68a^^  +  lUa^b^. 
Also,  (4a363  —   7c2(i^)2  = 

THEOREM    III. 

The  product  of  the  sum  of  two  quantities  multiplied  by  thetr 
difference,  is  equal  to  the  difference  of  their  squares. 

Let  the  quantities  be  denoted  by  a  and  b. 

Then,     a  -\-  b  z=.  their  sum,  and    a  —  6  =  their  difference 

We  have,  from  known  principles, 

{a-irb)x{a  —  b)  =  «2  _  ^2^ 
which  is  the  algebraic  enunciation  of  the  theorem. 

To  apply  this  principle  to  an  example,  we  have 

(8a3    +  7ai2)  x  (Sa^    —  lab"-)  =  64^^  _  49a2i*. 
Also,         (Vc  +  lah^)  X  {9a^c  —  lab^)  = 


34  KLEMENTS    OF    ALGEBRA.  [CHAP.   II 

47.  By  considering  the  last  three  results,  it  will  be  perceived 
that  their  composition,  or  the  manner  in  which  they  are  formed 
from  the  midtiplicand  and  multiplier,  is  entirely  independent  of 
any  particular  values  that  may  be  attributed  to  the  letters  a  and  b 
which  enter  the  two  factors. 

The  manner  in  which  an  algebraic  product  is  formed  from  its 
wo  factors,  is  called  the  law  of  this  product  ;  and  this  law  re- 
mains always  the  same,  whatever  values  may  be  attributed  to  the 
letters  which  enter  into  the  two  factors. 

Of  factoring  Pohpiomlals. 

48.  A  given  polynomial  may  often  be  resolved  into  two  factors 
by  mere  inspection.  This  is  generally  done  by  selecting  all  the 
factors  common  to  every  term  of  the  polynomial  for  one  factor, 
and  writing  what  remains  of  each  term  within  a  parenthesis  foi 
the  other  factor. 

1.  Take,  for  example,  the  polynomial 

ab  -j-  ac ; 
in  which,  it  is  plain,  that  a  is  a  factor  of  both  terms :    hence 
ah  -\-  ac  ■=  a  [b  -\-  c). 

2.  Take,  for  a  second   example,  the  polynomial 

ab'^-c  -}-  5oi^  "1-  ab'^c^. 

It  is  plain  that  a  and  Jf-  are  factors  of  all  the  terms  :    hence 

ab-^c  +  5aP  +  ab^"^  —  ab"^  (c  +  5i  +  c% 

3.  Take  the  polynomial  25a*  —  SOa^Z*  +  ISa^i^  .  jt  jg  evident 
that  5  and  d^  are  factors  of  each  of  the  terms.  We  may,  there 
fore,  put  the   polynomial  under  the  form 

5«2  (5a2  —  6«i  +  3^2). 

4.  Find  the   factors   of    "^a^  -}-  9o^c  +  ISa^o-y. 

Ans.  3a2  (5  -f  3c  +  6;ry). 

5.  Find  the  factors   of    Sa^ca:  —  \^acx^  +  2arc-^y  —  SOa'^c^x. 

Ans.  9.ac  {Aax  —  9x^  +  c*y  —  ISaVa-). 

6.  Find   the  factors  of    2ia%'^cx  —  ZOa%^c^y -\- 2(Sa^¥cd  +  (Jabc. 

Ans.  6abc  {iabx  -  baU^c^y  +  6a«^»V  +  1). 
7     Find  the   factors  of    a^  +  2ah  -f  b"^. 

Ans.  {a  +  h)  via  +  i) 


CHAP.  II.]  DIVISION.  35 

8.  Find  the  factors  of    a^  _  yi  Ans.  (a  +  6)  x  (a  —  V). 

9.  Find  the  factors  of    a-  —  lah  +  h"^. 

Ans.  (a  —  b)  X  (a  —  b). 

10.  Find   the    factors   of  the    poljTiomial     6a^b -}- Sa'^b^  —  I6ab'' 

-  2ab. 

11.  Find  the  factors  of  the  polynomial     ISaic^  —  3bc^  +  Qa^iV 

-  12f///c2. 

12.  Find    the    factors    of    the    polynomial      25a^bc^  —  30a^bc*d 
•  5ac*  —  60ac^. 

13.  Find  the  factors  of  the  polynomial    42a'^b'^  —  7 abed  +  labd 

Ans.  lab  (Joab  —  cd  -{-  d). 


DIVISION. 

49.  Division,  in  Algebra,  explains  the  method  of  finding  from 
iwo  given  quantities,  a  third  quantity,  which  multiplied  by  the 
first  shall  produce  the  second. 

The  first  of  the  given  quantities  is  called  the  divisor:  the  sec- 
ond, the  dividend;  and   the  third,  or    quantity  sought,  the  quotient. 

Let  U3  first  consider  the  case  of  two  monomials,  and  divide 
"ibaWc  by   lab. 

The  dinsion  may  be  indicated  thus, 

— -  =  5a^-^b'^-\^=  baHc^. 

lab 

Now,  since  the  quotient  must  be  such  a  quantity  as  multiplied 
by  the  divisor  will  produce  the  dividend,  the  co-efficient  of  the 
quotient  multiplied  by  7  must  give  35,  the  co-efficient  of  thv 
dividend ;  hence,  the  new  co-efficient  5  is  found  by  dividing  35 
by  7.  Again,  the  exponent  of  any  letter,  as  a,  in  the  quotient, 
added  to  the  exponent  of  the  same  letter  in  the  divisor,  must 
give  the  exponent  of  this  letter  in  the  dividend :  hence,  the  ex- 
ponent in  the  quotient  is  found  by  subtracting  the  exponent  in 
the  divisor  from  that  in  the   dividend.     Thus,  the   exponent  of 

a  is   3  —  1  —  2,    and  of  6,   2  —  1  =  1, 
and   since   c   is   not  found  in  the   divisor,  there    is    nothing  to   be 
subtracted  from  its  exponent. 

bO.  Hence,  for  the  division  of  monomials,  we  have  the  following' 


36  ELEMENTS    OF    ALGEBRA.  [CHAP.  11. 

RULE. 

I.  Divide  the  co-ejicient  of  the  dividend  by  the  co-ejicient  of  the 
divisor,  for  a  new  co-efficient. 

II.  Write  after  this  co-efficient,  all  the  letters  of  the  dividend, 
and  affect  each  with  an  exponent  equal  to  the  excess  of  its  exponent 
in   the   dividend  over  that  in  the  divisor. 

From  this  rule  we  find, 


ASa^bh'^d 


\2ab-^c 

1.  Divide     16a:' 

2.  Divide 


=  4a^bcd ; 


l50a^Pcd^ 
30a3<i2 


by    8x. 
15a'^xi/^   by    Say. 

3.  Divide    Siab^x   by    12^2. 

4.  Divide    96a*b'^c^   by    l2a^c. 

5.  Divide     I44a^^c''d=   by    360**6^6^^ 

6.  Divide    256a^c^^x^   by    Ida'^cx'^. 

7.  Divide    SOOa^^c^x'^   by    dOa^b^c^x. 


=  ba^lPcd. 

Ans.  2x 

Ans.  5axi/^. 

Ans.   7abx. 

Ans.  Sa^bc^. 

Ans.  4a^b^cd*. 

Ans.  IGubcx. 

Ans.  lOabcx. 


51.  It  follows  from  the  preceding  rule  that  the  exact  division 
of  monomials  will  be  impossible: 

1st.  When  the  co-elficient  of  the  dividend  is  not  divisible  by 
that  of  the  divisor. 

2d.  When  the  exponent  of  the  same  letter  is  greater  in  the 
divisor  than  inj»the  dividend. 

3d.  When  the  divisor  contains  one  or  more  letters  which  are 
not  found  in  the  dividend. 

When  either  of  these  three  cases  occurs,  the  quotient  remains 
under  the  form  of  a  monomial  fraction  ;  that  is,  a  monomial  ex- 
pression, necessarily  containing  the  algebraic  sign  of  division. 
Such  expressions   may  frequently  be  reduced. 

,„  ,        ^  ,  \2a*b^cd       Sa'^bd 

Take,  for  example,         _  „.  „    =  -— — 

Here  an  entire  monomial  cannot  be  obtained  for  a  quotient ; 
for,  12  is  not  divisible  by  8,  and  moreover,  the  exponent  of  c 
is  less  in  the  dividend  than  in  the  divisor.  But  the  expression 
can  be  reduced,  by  dividing  the  numerator  and  denominator  by 
the  factors  4,  a'^,  b,  and  c,  which  are  common  to  both  the  terms 
of  the  fraction. 


CHAP.  II.]  DIVISION.  37 

In  general,  to  reduce  a  monomial  fraction,  we  have  the  fol- 
lowing 

RULE. 

Suppress  all  the  factors  common  to  the  numerator  and  denomina- 
tor, and  write  those  letters  xohich  are  not  common,  with  their  respec- 
Itce  exponents,  in   the  term  of  the  fraction  which   contains  them. 

From  this   rule   we  find, 

ASa'^b'^cd^   _  Aad'^  37aPc^d    _  TtV^c  _ 

2,^d^hHHe  ~  ZhTe       ^^  %a%c'd'^  ~  Qd'd  ' 

\2a%^c^         Sab  ,        7a'^b  1 

also, = ;      and     = ; 

16a' b^c^         4c2  14^362       2ab  ' 

In  the  last  example,  as  all  the  factors  of  the  dividend  are 
found  in  the  divisor,  the  numerator  is  reduced  to  tmity ;  for,  in 
fact,  both  terms  of  the  fraction  are  divided  by  the  numerator. 

52.  It  often  happens,  that  the  exponents  of  certain  letters,  are 
the  same  in  the  dividend  and  divisor. 

2ia^b^ 
For  example,       - —  =  3a, 

is  a  case  in  which  the  letter  b  is  aflected  with  the  same  expo- 
nent in  the  dividend  and  divisor  :  hence,  it  will  divide  out,  and 
will  not  appear  in  the  quotient. 

But  if  it  is  desirable  to  preserve  the  trace  of  this  letter  in 
the  quotient,  we  may  apply  to  it  the  rule  for  the  exponents  (Art. 
50)  :    wliich  gives 

b^ 

—  =  62-2  ^  ^,0 

This  new  symbol  Z>'\  indicates  that  the  letter  b  enters  0  times 
as  a  factor  in  the  quotient  (Art.  13);  or  what  is  the  same  thing, 
that  it  does  not  enter  it  at  all.  Still,  the  notation  shows  that  b 
was  in  the  dividend  and  divisor  with  the  same  exponent,  and  has 
disappeared  by  division. 

1  5(1   DC 

In  like   manner,  =  5a%^c^  =  5&2. 

Sa^bc^ 

53.  We  will  now  show  that  the  power  of  any  quantity  whosy 
exponent  is  0,  is  equal  to  unity.  Let  the  quantity  be  represented 
jy  a,  and  let  7/1  denote   any  exponent  whatever 


38  ELEMENTS    OF    ALGEBRA.  [CHAP.  11 

Then,       —  =  a"'"'"  =  a",  by  the  nile  of  division. 

But,  —  =  1,  since  the  numerator  and  denominator  are  equal: 

hence,  o°   r=  1,  since  each  is  equal  to—. 

^  a"' 

We  observe  again,  that  the  symbol  a^  is  only  employed  con- 
ventionally, to  preserve  in  the  calculation  the  trace  of  a  letter 
which  entered  in  the  enunciation  of  a  question,  but  which  may 
disappear  by  division. 

Division  of  Polynomials. 

54.  The  object  of  division,  is  to  find  a  third  polynomial  called 
the  quotient,  w^hich,  multiplied  by  the  divisor,  shall  produce  the 
dividend. 

Hence,  the  dividend  is  the  assemblage,  after  reduction,  of  the 
partial  products  of  each  term  of  the  divisor  by  each  term  of  the 
quotient,  and  consequently,  the  signs  of  the  terms  in  the  quotient 
must   be    such    as  to    give  proper    sign^    to    the  partial  products. 

Since,  in  multiplication,  the  product  of  two  terms  having  the 
same  sign  is  affected  with  the  sign  +,  and  the  product  of  two 
terms  having  contrary  signs,  with  the  sign    — ,  we  may  conclude, 

1st.  That  when  the  term  of  the  dividend  has  the  sign  +»  and 
that  of  the  divisor  the  sign  +,  the  term  of  the  quotient  must 
have  the  sign  +. 

2d.  When  the  term  of  the  dividend  has  the  sign  +,  and  that 
of  the  divisor  the  sign  — ,  the  term  of  the  quotient  must  have 
the  sign  —  ;  because  it  is  only  the  sign  — ,  which,  combined  with 
the  sign   — ,   can  produce  the  sign  +    of  the  dividend. 

3d.  When  the  term  of  the  dividend  has  the  sign  — ,  and  that 
of  the  divisor  the   sign   +,  the  quotient  must  have  the  sign   — . 

That  is,  when  the  two  corresponding  terms  of  the  dividend 
and  divisor  have  the  same  sign,  their  quotient  will  be  affected 
with  the  sign  +>  and  when  they  are  affected  with  contrary  signs, 
their  quotient  will  be  affected  with  the  sign  —  ;  again,  for  the 
sake  of  brevity,  we   say  that 

+   divided  by  +,  and  —   divided  by   — ,  give   -f  ; 
—   divided  by   +,  and   +    divided  by   — ,  give    — • 


CHAP.  II.] 


39 


Dividend. 

Divisor. 

a2 

g2 

— 

2ax 
ax 

+ 

x' 
a;2 

IL 

a  —  X 

a  —  X 

— 

ax 

Quotient. 

— 

ax 

+ 

x2. 

FIRST    EXAMPLK 

Divide    a^  —  2ax  +   x^    by    a  —  x 

It  is  found  most  convenient, 
in  division  in  algebra,  to  place 
the  divisor  on  the  right  of  the 
dividend  and  the  quotient  di- 
rectly under  the  divisor. 

We  first  divide  the  term  a"^  of  the  dividend  by  the  term  a  of 
the  divisor,  the  partial  quotient  is  a,  which  we  place  under  the 
divisor.  We  then  multiply  the  divisor  by  a,  and  subtract  the 
product  a-  —  ax  from  the  dividend,  and  to  the  remainder  bring 
down  x"^.  We  then  divide  the  first  term  of  the  remainder,  —  ax, 
by  a,  the  quotient  is  —  x.  We  then  multiply  the  divisor  by 
—  X,  and,  subtracting  as  before,  we  find  nothing  remains.  Hence, 
a  —  X  is  the  exact  quotient. 

SECOND    EXAMPLE. 

Let  it  be  required  to  divide  26^2^2  ^  lOa*  —  4Sa'%  +  24uP  by 
4ab  —  5a^  -f  3P.  In  order  that  we  may  follow  the  steps  of  the 
operation  more  easily,  we  will  arrange  the  quantities  with  refer- 
ence to  the  letter  a. 


Dividend. 
10a*  —  480^6  +  2Ga"^2  _|_  24aP 

-f  lOo*  —     8a^  —     &aW 

—  40a3r4-~32a2i2  _^  ^Aa^ 

—  40a^  +  32a2Z/'-  -f  24aP. 


Divisor. 
5a2  +  4ab  +  3^2 


—  2a2  -I-  8ab 
Quotient . 


It  follows  from  the  definition  of  division,  and  the  rule  for  the 
multiplication  of  polynomials  (Art.  44),  that  the  dividend  is  the 
assemblage,  after  addition  and  reduction,  of  the  partial  products 
of  each  term  of  the  divisor,  by  each  term  of  the  quotient  sought. 
Hence,  if  we  could  discover  a  term  in  the  dividend  which  was 
derived,  without  reduction,  from  the  multiplication  of  a  term  of 
the  divisor  by  a  term  of  the  quotient,  then,  by  dividing  this  term 
of  the  dividend  by  that  of  the  divisor,  we  should  obtain  a  term  of 
the  required  quotient. 

Now,  from  the  third  remark,  of  Art.  45,  the  term  lOa"",  affected 
with  the  highest  exponent  of  the  letter  a,  is  derived,  without  re- 
duction  from  the  two  terms   of  the    divisor   and   quotient,   afl'ected 


40  ELEMENTS    OF    ALGEBRA.  [CHAP.  11. 

with  the  highest  exponent  of  the  same  letter.  Hence,  by  dividing 
the  term  10a*  by  the  term  —  5a^,  we  shall  have  a  term  of  the 
required  quotient. 

Dividend.  Divisor. 

10a*  —  ASa^b  +  2&aW  +  2AaP 
+  10a*  —     Sa^b  —    6aW 


5a2  -(-.  4ab  +  3b^ 


—  40a^  +  32a%^  +  2-iab^ 

—  40a^  +  'i2aW  +  2\ab'^. 


—  2a2  +  8 
Quotient. 


Since  the  terms  10a*  and  —  5a-  are  affected  with  contrary 
signs,  their  quotient  will  have  the  sign  —  ;  hence,  10a*,  divi- 
ded by   —  ott^,   gives  —  20^  for   a  term  of  the  required  quotient. 

After  having  written  this  term  under  the  divisor,  multiply  each 
term  of  the  divisor  by  it,  and  subtract  the  product, 

10a*  —  8a^  -h  6a^^, 

from  the   dividend,   which  is   done   by  writing    it  below  the   divi- 
dend, conceiving  the  signs  to  be  changed,   and  performing  the  re- 
duction.    Thus,  the   remainder  after  the   first  partial   division  is 
—  40a^  +  32a2^>2  +  24ab\ 

This  result  is  composed  of  the  partial  products  of  each  term 
of  the  divisor,  by  all  the  terms  of  the  quotient  which  remain  to 
be  determined.  We  may  then  consider  it  as  a  new  dividend,  and 
reason  upon  it  as  upon  the  proposed  dividend.  We  will  there- 
fore divide  the  term  —  iOa^^b,  affected  with  the  highest  exponent 
of  a,  by  the  term  —  Sa^  of  the  divisor.  Now,  from  the  prece- 
ding principles, 

—  40a^,  divided  by  —  Sa^,  gives  -f  8a5 
for  a  new  term  of  the  quotient,  which  is  written  on  the  right  of 
the  first.  Multiplying  each  term  of  the  divisor  by  this  term  of 
the  quotient,  and  writing  the  products  underneath  the  second  divi- 
dend, and  making  the  subtraction,  we  find  that  nothing  remains. 
Hence. 

—  2a-  4-  Sab    or    Sab  —  2a^ 

IS   the  required  quotient,   and   if  the   divisor   be    multiplied   by  it. 
f'le   product  will  be   the  given  dividend. 

Hy  considering  the  preceding  reasoning,  we  see  that,  in  each 
partial    operation,   we   dinde    that   term  of  the   dividend  which   is 


CHAP,   n.]  DIVISION.  4l 

affected  with  the  highest  exponent  of  one  of  the  letters,  by  that 
term  of  the  divisor  affected  with  the  highest  exponent  of  the 
same  letter.  Now,  we  avoid  the  trouble  of  looking  out  these 
terms  by  writing,  in  the  first  place,  the  terms  of  the  dividend  and 
divisor  in  such  a  manner  that  the  exponents  of  the  same  letter  shall 
go   on   diminishing  from  left  to  right. 

This  is  Avhat  is  called  arranging  the  dividend  and  divisor  with 
reference  to  a  certain  letter.  By  this  preparation,  the  first  term 
on  the  left  of  the  dividend,  and  the  first  on  the  left  of  the  divisor, 
are  always  the  two  which  must  be  divided  by  each  other  in  or- 
der to  obtain  a  term  of  the  quotient. 

55.  Hence,  for  the  division  of  polynomials  we  have  the  fol- 
lowing 

RULE. 

I.  Arranore  the  dividend  and  divisor  with  reference  to  a  certain 
letter,  and  then  divide  the  first  term  on  the  hft  of  the  dividend  by 
the  first  term  on  the  left  of  the  divisor,  for  the  first  term  of  the 
quotient ;  muliply  the  divisor  by  this  term  and  subtract  the  prod- 
uct from  the  dividend. 

II.  Then  divide  the  first  term  of  the  remainder  by  the  first  term 
of  the  divisor,  for  the  second  term  of  the  quotient ;  multiply  the 
divisor  by  this  second  term,  and  subtract  the  product  from  the  re- 
sult of  the  first  operation.  Continue  the  same  process,  and  if  the 
remainder  is  0,  tlie  division  is  said   to  be   exact. 

THIRD    EXAMPLE. 

Divide  21a;3y2  +  25xh/  +  68^3/*  —  40y5  —  bQx^  —  IQx^y  by 
5y^  —  S-r^  —  Qxy. 

—  40y5  -f  68a:y*  +  25a;2y3  +  2lx^y'^  —  IQx^y  —  56*5  llSy^  —  Qxy  —  8x^ 

—  40y'  +  48.ry-t  -f  64x^^3  _  g^g  ^  ^^^^  _  3^,2^  4.  7^^ 

1st  rem     20xy*  —  SOx^y^  +  2 1  xhf 
20.Ty*  —  24a:2y3  _  32a:^y2 

2d  rem.     ~^^      —  1 5x2y3  +  53^^  —  18a-*y 
—  15x2y3  ^  iSxy-  +  24jiy 


3d  rem.         ...       -    '3oj-^y^ —  42x*y  —  56x^ 

35a:3y2  —  42x*y  —  56a^ 

Final  remainder  -       -       .       -     0 


42  KLKMENTS     OK    ALOKBRA.  [CHAP.   11. 

56.  Remark. — In  performing  the  division,  it  is  not  necessar}- 
to  bring  down  all  the  terms  of  the  dividend  to  Ibrin  the  first  re- 
mainder, but  they  may  be  brought  down  in  succession,  as  in  tlie 
example. 

As  it  is  important  that  begiimers  should  render  themselves 
familiar  with  the  algebraic  operation,  and  acquire  the  habit  of 
calculating  promptly,  we  will  treat  this  last  example  in  a  different 
manner,  at  the  same  time  indicating  the  simplifications  which 
should  be  introduced.  These,  consist  in  subtracting  eacli  jiartial 
product  from  the   dividend  as   soon  as   this  product   is   formed. 

—  40 If  +  68xf  +  •35a:-y3  -|-  2 1  x^f  —  1 8x^y  —  56af ^  1 1 5f  —  6xy  —  So-^ 
1st  rem.    20.ry^  —  39jY  +  2\xY  —  8f  +  Axf  —  3x^y  -f  Ta^ 

2d  rem.        -      —  Ibx'^f  +  53a:^y-  —  ISar^y 
3d  rem.        -  -       -  SSjr^y^  —  A2x^y  —  56a;^ 


Final  remainder    -       -       -       -       -  0. 

First,  by  dividing  —  40y^  by  Sy^,  we  obtain  —  Sy^  for  the  quo- 
tient. Multiplying  Sy^  by  —  8y?,  we  have  —  AOy^,  or  by  chan- 
ging the  sign,  +  40y^,  wliich  destroys  the  first  term  of  the  divi- 
dend. 

In  like  maimer,  —  Qxy  X  —  Sy-'  gives  +  48ary*,  and  for  the 
subtraction  —  4S.ry'',  which  reduced  with  +  QQxy^,  gives  20a:y* 
for  a  remainder.  Again,  —  8x!^  x  —  Sy^  gives  +,  and  changing 
the  sign,  —  QAx'^y'^,  which  reduced  with  2ox'^y^,  gives  —  39x^y^ 
Hence,  the  result  of  the  first  operation  is  20a;y*  —  39a;^y^,  fol- 
lowed by  those  terms  of  the  dividend  which  have  not  been  re- 
duced with  the  partial  products  already  obtained.  For  the  sec- 
ond part  of  the  operation,  it  is  only  necessary  to  bring  down  the 
next  term  of  the  dividend,  to  separate  this  new  dividend  from  the 
primitive  by  a  line,  and  to  operate  upon  this  new  dividend  in 
the  same  manner  as   we  operated  upon  the  primitive,  and  so  on. 

FOURTH    EXAMPLE. 

Divide  -  -  95a  —  73«2  -f-  560*  —  25  —  59a3  by  —  3u^ 
-1-5  —  11a  —  7a3. 


56a^  —  59a3  —  73a2  +  95a  —  25 


ist  rem.      —  35a3  -f  15a2  +  55a  —  25 


7«3_  3a2  _  Ua  _^  5 


8a   —  5 


2d  remainder     -       -       0. 


CHAP.  11.]  DIVISION.  43 

GENERAL    EXAMPLES. 

1.  Divide  lOab  +  loac   b}'    5a.  Ans.  2b  +  3c. 

2.  Divide  30ax  —  54x   by    6x.  Ans.  5a  —  9. 

3.  Divide  lOx'^y  —  \5y^  —  5y   by   5y.  Ans.  2x^  —  2y —  1. 

4.  Divide  12a  +  'iax  —  ISax^   by    3a.  Ans.  4  +  x  —  &x^. 

5.  Divide  Qax^  -\-  9a"x  +  a^^^    by    ax.  Ans.  6x  +  ''^a  -j-  aoc. 
G.  Divide  a-  +  2aj?  -4-  x^   by    a  +  a:.  A/i^.  a  -}-  a-. 

7.  Divide     a^  —  3a~y  +  Say^  —  y^    by    a  —  y. 

Ans.   c^  —  2ay  4"  v^ 

8.  Divide    24a2i  —  VlaHlP-  —  6ab    by    —  6ab. 

Ans.   -  Aa  +2aHb  +  1. 

9.  Divide    6x<— 96   by  3x— 6.     Ans.  2x^ -\- Ax- +  Qx  +  \Q. 

10.  Divide       -       -       a^  —  5a^x  +  lOu^x"^  —  lOa^x^  +  Sox*  —  x^ 
by    a^  —  2aa:  +  x"^.  Ans.  a^  —  Sa'^x  +  Saj;^  —  x^ 

11.  Divide    48x3  _  75^:^2  —  64a2x  +  lOSa^   by   2x  —  3a. 

Ans.  24x2  —  2ax  —  350^. 

12.  Divide    y^  —  Sy^x^  -f  Sy^x*  —  x^    by   y^  —  3y'^x  -f-  Syx^  —  x^, 

Ans.  y^  -f  Sy^x  +  Syx^  -j-  x^. 

13.  Dinde     %\a^b^ —25a'^h^     by    ^aW  ^  5abK 

Ans.  8a^^  —  5ab\ 

14.  Divide     6a^  +  23a2^  +  22ai2  4.  5P    by    3a2  +  4a^'  +  b^. 

Ans.  2a  -j-  5b. 

15.  DiA'ide     6ax^  +  6ax^y^  -{-  A2a'^x'^    by    ox  +  5ax. 

Ans.  x^  +  xy*^  +  7ax. 

16.  Divide     —  ISa"  +  37a2iJ  —  29a''-cf  —  20b^<P  +  AAhcdf—  Sc^-* 
by    3a2  —  5bd  +  c/  Ans.    —  Sa^  +  46J  —  8cf. 

17.  Divide    x*  +  x^y^  -\-  y*   by   x^  —  xy  +  y^, 

Ans.  x"^  -{-  xy  -{-  y"^. 

18.  Divide    x*  —  y*    by   x  —  y.  Ans.  x^  +  ^^y  +  ^y^  +  y"'- 

19.  Divide     3a*  —  QaW  +  3a2c2  +  5i*  —  3/^2^2    by    a2  —  ^2. 

Ans.  3a2  —  5^-2  +  3c2. 

20.  Divide   6x6—  5x-^y2_  63.4^4  _|_  Q^2y2  _|_  I5a:3y3  _  9^2y4  ^  j  Ox^y^ 
f  15y5   by    3x3  4.  2x2/  _|_  3^2,  ^„^.  2x3  _  33.2^2  _^  5^3 


44  ELEME.VTS    OF    ALGEBRA.  [CHAP.  II. 

RemarJcs  on  the  Division  of  Folynomials. 

57.  When  the  first  term  of  the  arranged  dividend  is  not  ex- 
actly divisible  by  that  of  the  arranged  divisor,  the  complete  divis- 
ion is  impossible  ;  that  is  to  say,  there  is  not  a  polynomial 
which,  multiplied  by  the  divisor,  will  produce  the  dividend.  And 
ill  general,  we  shall  find  that  a  division  is  impossible,  wheii  the 
first  term  of  any  one  of  the  partial  dividends  is  not  divisible  by 
the  frst   term  of  the  divisor. 

We  Avill  add,  as  to  polynomials,  that  it  may  often  be  discov- 
ered by  mere  inspection  that  they  are  not  divisible.  When  the 
polynomials  contain  two  or  more  letters,  observe  the  two  terms 
of  the  dividend  and  divisor,  Avhich  are  affected  with  the  highest 
exponent  of  each  of  the  letters.  If  these  terms  do  not  give  an 
exact  quotient,  we  may  conclude  that  the  total  division  is  im- 
possible. 

Take,  for  example, 

12a3  -  ba'b  +  7a52  -  IW  \\^a^  -\-  Bab -\-  Sb"^ 

By  considering  only  the  letter  a,  the  division  would  appear 
possible  ;  but  regarding  the  letter  b,  the  division  is  impossible, 
S'nce    —  llb^   is  not  divisible  by  Si^. 

58.  One  polynomial  ^4,  cannot  be  divided  by  another  B  con- 
taining a  letter  which  is  not  found  in  the  dividend ;  for,  it  is 
impossible  that  a  third  quantity,  multiplied  by  B  which  contains 
a  certain  letter,  should   give  a  product  independent  of  that  letter. 

A  monomial  is  never  divisible  by  a  polynomial,  because  every 
polynomial  multiplied  by  either  a  monomial  or  a  polynomial  gives 
a  product  containing  at  least  two  terms  which  are  not  suscep- 
tible of  reduction. 

59.  If  the  letter,  with  reference  to  which  the  dividend  is  ar- 
ranged, is  not  found  in  the  divisor,  the  divisor  is  said  to  be  in- 
dependent of  that  letter ;  and  in  that  case,  the  exact  division  is 
impossible,  unless  the  divisor  will  divide  separately  the  co-ejicient 
of  the  leading  letter  of  each  term  of  the  dividend. 

For  example,  if  the   dividend  were 

3&a*  +  95a2  +  12i, 
arranged  with  reference  to  the    letter  a,  and  the    divisor    35,  the 


CHAP.  II.]  DIVISION".  45 

divisor  would  be  independent  of  the  letter  a ;  and  it  is  evident  thai 
the  exact  division  could  not  be  performed  unless  the  co-efficient 
of  each  term  of  the  dividend  w^ere  divisible  by  3i.  The  expo- 
nents of  the  leading  letter  in  the  quotient  would  be  the  same  as 
in  the   dividend. 

1.  Divide     ISa'^X^  —  ^Qa^x"^  —  \2ax    by    6a:. 

Ans.   Sa^x  —  6a2x2  —  2a. 

2.  Divide    25a*i  —  SOa^i  +  AOah   by  5b. 

Ans.  5a*  —  60^  -)_  8a. 

60.  Although  there  is  some  analogy  between  arithmetical  and 
algebraical  division,  with  respect  to  the  manner  in  which  the 
operations  are  disposed  and  performed,  yet  there  is  this  essential 
difference  between  them,  that  in  arithmetical  division  the  figures 
of  the  quotient  are  obtained  by  trial,  while  in  algebraical  division 
the  quotient  obtained  by  dividing  the  first  term  of  the  partial  divi- 
dend by  the  first  term  of  the  divisor,  is  always  one  of  the  terms 
of  the  quotient  sought. 

From  the  third  remark  of  Art.  45,  it  appears  that  the  term  of 
the  dividend  afiected  with  the  highest  exponent  of  the  leading 
letter,  and  the  term  afiected  with  the  lowest  exponent  of  the 
same  letter,  may  each  be  derived  without  reduction,  from  the 
multiplication  of  a  term  of  the  divisor  by  a  term  of  the  quotient. 
Therefore,  nothing  prevents  our  commencing  the  operation  at  the 
right  instead  of  the  left,  since  it  might  be  performed  upon  the 
terms  afiected  with  the  lowest  exponent  of  the  letter,  with  ref- 
erence  to  which  the  arrangement  has  been  made. 

Lastly,  so  independent  are  the  partial  operations  required  by 
the  process,  that  after  having  subtracted  the  product  of  the  di- 
visor by  the  first  term  found  in  the  quotient,  we  could  obtain 
another  term  of  the  quotient  by  dividing  by  each  other  the  two 
terms  of  the  new  dividend  and  divisor,  afiected  with  the  highest 
exponent  of  a  different  letter  from  the  one  first  selected.  If  the 
same  letter  is  preserved,  it  is  only  because  there  is  no  reason 
for  changing  it,  and  because  the  two  poljnaomials  are  already 
arranged  with  reference  to  it;  the  first  terms  on  the  left  of  \\w 
dividend  and  divisor  being  sufficient  to  obtain  a  term  of  tlu^ 
quotient ;  whereas,  if  the  letter  is  changed,  it  becomes  necessary  to 
seek  for  the  highest  exponent  of  the  new  letter. 


46  ELEMENTS    OF    ALGEBRA.  [CHAP.    H 

61.  Among  the   different   examples   of   algebraic    division,  'tliere 
is   one   remarkable    for  its    applications.     It    is   expressed  thus : 

The  difference   between   the  same  powers  of  any  two  quantities  is 
alirays   divisible   bi/   the   difference   between   the  quantities. 

Let  the  quantities  be   represented  by   a  and  b ;    and  let  m  de- 
note any  positive   whole  number.     Then, 

gm  _  Jm 

will  express  the  difference  between  the  same  powers  of  a  and  b, 
and  it  is  to  be  proved  that  a"  —  b^  is  exactly  divisible  by  a  —  b. 
If  we  begin  the   division    of   a'"  —  Z*""   by    a  —  b,    we  have 


a'^-^b 


a  —  b 


1st  rem.         .       -       -       .       a"'~'^b  —  b"^ 
or,  by  factoring   •       -       -    b  (a"*"'  —  5'""^). 

Dividing  a'"  by  a  the  quotient  is  a'"-!,  by  the  rule  for  the  ex- 
ponents. The  product  of  a  —  J  by  a"""'  being  subtracted  from 
the  dividend,  the  lirst  remainder  is  a^~^b  —  b'",  which  can  be 
put  under  the  form    h  (a"'-^  —  &"»"'). 

Now,  if  the  factor  (fl"*-!  —  b'"-^)  of  the  remainder,  be  di^nsi- 
ble  by  a  —  b,  it  follows  that  the  dividend  a'"  —  b'"  is  also  divisi- 
ble by  a  —  b  :    that   is, 

If  the  difference  of  the  same  powers  of  two  quantities  be  diinsi- 
hle  by  the  difference  of  the  quantities,  then,  the  difference  of  the 
powers  of  a   degree  greater  by   unity   is  also   divisible   by   it. 

But  by  the  rules  for  division,  we  have 

a2  -  i2 

J-  =a  +  b. 

a    —  0 

Hence,  we  know,  from  Avhat  has  just  been  proved,  that  a^  ~  b^ 
is  divisible  by  a  —  b,  and  from  that  result  we  conclude  that 
a*  —  b*  is  divisible  by  a  —  b,  and  so  on,  until  we  reach  any 
exponent  at  rn. 


IHAP.   III. I  ALGEBRAIC    FRACTIONS. 


CHAPTER  III. 


OF    ALGEBRAIC    FRACTION'S. 


62.  Algf,braic  fractions  are  to  be  considered  in  the  .  same 
point  of  view  as  arithmetical  fractions  ;  that  is,  a  unit  is  sup- 
posed to  be  divided  into  as  many  equal  parts  as  there  are  units  in 
the  denominator,  and  one  of  these  parts  is  supposed  to  he  taken 
as  many  times  as   there  are  units  in  the  numerator. 

Thus,  in  the  fractional  expression 

a^h 
c  -\-d' 

a  given  imit  is  supposed  to  be  divided  into  as  many  equal  parts 
as  there  are  units  in  c  -{-  d,  and  as  many  of  these  parts  are 
taken,  as  there  are  units  in  a  -\-  b. 

The  rules  for  performing  Addition,  Subtraction,  Multiplication, 
and  Division,  are  the  same  as  in  arithmetical  fractions.  Hence, 
it  will  not  be  necessary  to  demonstrate  these  rules,  and  in  their 
application  we  must  follow  the  methods  already  indicated  in  sim- 
ilar operations  on   entire   algebraic  quantities. 

63.  Every  quantity  which  is  not  expressed  under  a  fractional 
form,  is  called  an  entire  algebraic  quantity. 

64.  An  algebraic  expression,  composed  partly  of  an  entire 
quantity  and  partly  of  a  fraction,  is  called  a  mixed  quantity. 

65.  When  the  division  of  two  monomial  quantities  cannot  be 
performed  exactly,  it  is  indicated  by  means  of  the  known  sign, 
and  in  this  case,  the  quotient  is  presented  under  the  form  of  a 
fraction,  which  we  have  already  learned  how  to  simplify   (Art.  51). 

With'  respect  to  polynomial  fractions,  the  following  are  cases 
which  are  easily  reduced. 


■iS  ELEMENTS    OF    ALGEBRA.  (CHAP.  HI. 

a'^  —  J2 


Take,  for  example,  the  expression 


a^  —  2ab  +  b" 


This    fraction    can  take   the   form    ) —f^\ ~     (Art.  46). 

[a  —  b)[a  —  0) 

Suppressing  the  factor  a  —  b.  which  is  common  to  the  two  terms, 

a  +  b 


we  obtain 

Again,  take  the   expression 


a  —  b 

5a3  —  lOa^-\-  5ab^ 


8a3  —  Sa'^b 

which  can  be  put  imder  the  form  (Art.  48) : 
5a  (a2  _  2ab  +  b^) 


which  is  equal  to 


8a^  {a  -  b) 
5a  {a  —  bf 


8«2  {a-b)' 

and  by  suppressing  the  common  factors,    a  {a  —  h),    the  result  is 

5{a  —  b) 
8^        ' 

In  the  particular  cases  examined  above,  the  two  terms  of  the 
fraction  are  decomposed  into  factors,  and  then  the  factors  com- 
mon to  the  numerator  and  denominator  are  cancelled.  Practice 
teaches  the  manner  of  performing  these  decompositions,  when 
they  are  possible. 

But  the  two  terms  of  the  fraction  may  be  complicated  poly- 
nomials, and  then,  their  decomposition  into  factors  not  being  so 
easy,  we  have  recourse  to  the  process  for  finding  the  greatest 
common  divisor,  which  is  explained  at  page   300. 

CASE  I. 
70.  To  reduce  a  fraction  to  its  simplest  form. 

RULE. 

I.  Decompose  the  numerator  and  denominator  into  factors,  as  in 
Art.  48. 

II.  Then  cancel  the  factors  common  to  the  numerator  and  de- 
nominator, and  the  result  will  be  the  simplest  form  of  the  fraction. 


CHAP.  III.)  ALGEBRAIC    FRACTIONS.  49 


EXAMPLES. 


_,     ,  ,         -         .  3fl6   +  6aC  .  .        ,  - 

1.  Reduce  the  fraction    — — ; — ■    to  its  simplest  fonn. 

3aa  4-  12a 

We  see,  by  inspection,  that   3    and   a   are   factors  of  the   nu- 
merator, hence 

3ab  -f  6ac  =  3a{b-\-  2c) 

We  also  see,  that  3  and  a  are  factors  of  the  denominator,  hence 

3ad-\-  12a  =  3a{d-^  4) 

3ab  +  6ac  _  3a  {b  +  2c)        b  +  2c 


Hence, 


3ad-i-12a        3a  {d  +  4)  d+4 


^    „   ^  Ga^  +  3ac  .        .      ,        , 

2.  Reduce     -r—, — -—r — ,    to  its  simplest  form. 
9ab   -f-  Sad 

2ab+c 
Ans. 


„    _    ,  25bc  +  5bf         .      , 

3.  Reduce     —-7- — ^    to  its  lowest  terms. 

Soo^  -|-  156 


Ans. 


36  +  d 

5c  +  J 
76  +  3" 


.     Ti    1  5iabc  .        .      ,         ^ 

4.  Reduce — ;    to  its  simplest  form. 

45a2c  -f-  9acd  ^ 

Ans 


^    _,    ,  36a^  -f  12a6/  .        .      ,     .   . 

5.  Reduce — — — —    to  Us  simplest  form. 

84a62  ^ 


Ans. 


5a  +  d' 

Sa-\-f 
7b 


\2acd 4cd^ 

6.  Reduce     7-; —7    to  its  simplest  form. 

I2cdf  +  AcH  ^ 

3a-  d 
Ans.  -— . 

3/-t-  c 

6or  —  /* 


_    T-,    ,  ISa^c^  —  3acf  .        .      ,        . 

7.  Reduce     —    to  its  simplest  form, 

27ac2    —  ^a(^  ^ 


Ans. 


9c  —  ^r^' 


CASE  n. 

71.  To  reduce  a  mixed  quantity  to  the  form  of  a  fraction. 

4 


M  ELEMENTS    OF    ALGEBRA.  ICHAP.  IV. 

RULE. 

Multiply  the  entire  part  hy  the  denominator  of  the  fraction  :  then 
conitfct  this  product  with  the  terms  of  the  numerator  by  the  rules 
fur  addition,  and  under  the   result  place  the  given   denominator 

EXAMPLES. 

1 .  Keduce     x  — to  the  form  of  a  fraction. 

X 

cP-  —  x"^       X-  —  (a^  —  x"^)       2x'  —  a^ 
Ans.  X = = . 

XXX 

ax  ~l~  X 

2.  Reduce     X to  the  form    of  a  fraction. 

2  a 


ax  —  ar^ 
Ans. 


2a 


2.7- 7 

3.   Reduce     5  -\ — '■ to  the  form  of  a  fraction. 

3a; 


17x- 
Ans. 


dx 

jp  d  —  \ 

4.  Reduce     1 to  the  form  of  a  fraction. 

a 

2a  —  X  -\-  1 

a 
J. 3 

5.  Reduce     1  -f  2j; to  the  form  of  a  fraction. 

5x 

10x2  +  4x-f-  3 

Ans. 

ox 

X  ~f~  a 

6.  Rtduce     3x  —  1 to  the  form  of  a  fraction. 

3a  —  2 

9ax  —  4a  —  7x  +  2 
An...  3-—^ 

CASE  III. 
7:1.  To  ro(Uice  a  fraction  to  an  entire,  or  mixed  quantity. 

RULE. 
Divide  the  numerator  hy  the  denominator  for  the  entire  part,  and 
/i/iice  the   remainder,  if  any    over  the  denominator  for  the  fractional 
fxirt. 


ax 

— 

x^ 

X 

ah 

— 

2a2 

CHAP,   ill.]  ALGEBRAIC    FRACTIONS.  51 

EXAMPLES. 

ax  -\-  a"^ 

1.  Reduce     to  a  mixed  quantity. 

X 

ax  -\-  a^                 c? 
Ans.   =  a  H • 

X  X 

2.  Keduce to  an  entire  or  mixed  quantity. 

Ans.  a  —  X. 

3.  Reduce     — — ^H.    to  a  mixed  quantity. 
b 

2a2 
Ans.  a 7—. 

0 

4.  Reduce     to  an  entire  quantity. 

a  —  X 

Ans.  a  -\-  X 

fjfyj  7v3 

5.  Reduce     —    to  an   entire  quantity. 

Ans.  x^  -\-  xy  -{-  y^. 

^    „    ,  10a:2  — 5a;+3  .      . 

6.  Keduce to  a  mixed  quantity. 

5a; 

3 

Ans.  2x  —  I  -\ . 

5.r 

CASE  IV. 

73    To  reduce  fractions  having  different  denominators  to  equiv 

alont  fractions  having  a  common  denominator. 

RULE. 
Multiply  each  numerator  into  all  the  denominators  except  its  oton, 
for   the  new   numerators,    and   all    the   denominators   together  for  a 
common  denominator. 

EXAMPLES. 

1.  Reduce    —    and    —    to  equivalent  fractions  having  a  com- 
mon denominator. 

the  new^  numerators. 


a  X  c  =:  ac  ^ 


and        -       b  X  c  =  be       the  common  denominator. 
2.  Reduce    —    and   to  fractions  having   a    common  de- 

.        ac         .    ab  -\-  h"^ 

nominator.  Ans.  7—    and    — . 

be  be 


52  ELEMENTS    OF    ALGEBRA.  [CHAP.  Ill 

3x     2b 

3.  Reduce    — ,    — ,    and  d,  to  fractions  having  a  common  de- 

2a      3c 

.         9cx       4ab  6acd 

nominator.  Ans.  - — ,     - — ,   and    — — . 

bac       oac  oac 

3       2x  2x 

4.  Reduce     — ,     — ,    and    a  -\ ,    to  fractions  having  a  coni- 

9a       Sax         ,     12a^  +  24ar 

mon  denommator.  Ans.  ——-,     - — ,   and   . 

12a'      12a'  12a 

1  (1  (2     -i—  Qc 

6.  Reduce    — ,     — ,    and    ,    to  fractions  having  a  com- 

mon  denominator. 

3a  +  3a:      2a^  -f  '^a^x  ,     6a^  +  6a;2 

Ans. — -,     -— — ,    and — . 

6a  -\-  ox         6a  -\-  ox  6a  -f-  6x 

6.  Reduce     -,     ,    and    — ,  to  fractions  having  a  com 

a  —  0         ax  c 

mon  denominator. 

a'^cx  ac^  —  ahc  —  hc^  +  ch^      +  a^bx  —  ah'^x 


Ans. 


obex  d^cx  —  abcx  a^cx  —  abcx 


CASE  V. 
74.  To  add  fractional  quantities  together. 

RULE. 

Reduce  the  fractions,  if  necessary,  to  a  common  denominator  : 
then  add  the  numerators  together  and  place  their  sum  over  the 
common  denominator. 

EXAMPLES. 

1,  Fmd  the  sum  of    -^,     —r,    and    —^. 

b        d  f 

Here,    -       a  X  d  xf  =  adf  ^ 

c  X  b  X  f  =  cbf  >  the  new  numerators. 

e  X  b  X  d  =  ebd  ) 
And       -       b  X  d  X  f  =  bdf      the  common  denominator. 
-_  adf        cbf        ebd        adf  -\-  cbf-\-  ebd 

2.  To    a-  ^    add    b  +  ^. 

0  c 

.  .    ,    .    2abx  —  Scjr' 

Ans.  a  -{-  0  A ; . 

be 


^  Ji  -4^  -^  ■  r  -r::,, 


0 


'•-\ 


CHAP.  III.]  ALGEBRAIC    FRACTIONS.  53 

3.  Add     — ,     — ,    and    —    together.  Ans.  a;  +  — -. 

2        3  4  12 

..,,«  — 2  4a;  19.r  —  14 

4.  Add     — - —     and    — -    together.  Ans. . 

3  7"  21 

.5.  Add     X  H to    3a:  -\ . 

3  4 

Ans.  4a:  -\ . 

12 

03!?  ST      I      // 

6.  It  IS  required  to  add     4a?,    ,    and    toirether. 

.         ^      ,    bx^  -{-  ax  -\-  o^ 

Ans.  4x  -i . 

2fla; 

2*      7a!  2a;  4-  1 

7.  It  is  required  to  add     — ^,     — ,    and    together. 

19.r  +  12 

Ans.  2x  4 . 

^         60 

8.  It  is  required  to  add     4ar,    — ,     and    2  -\ togeiiier. 

y  o 

.         ^      ,    4-la-  +  90 

Ans.  4x  A . 

^         45 

2x  8a; 

9.  It  is  required  to  add     3a;  H and    x —    together. 

23r 

Ans.  3x  -\ . 

45 

10.  What  is  the  sum  of    -,     -,    and 


Ans. 


a  —  b[     a  -\-  b'  a  -^  x' 

cfi  —  ax^  +  cP-h  —  hx^  +  «^<^  +  <i<^^  —  fl^c  — hex  +  cP-d  —  IPtl 
a?  —  IP-a  -\-  oPx  —  b'^x 
cP  -\-  cP  {I3  -\-  c  ■\-  A")  —  a  (a;2  —  ex  ■\-  he)  —  h  [ofl  -\-  ex  -\-  hd) 
a^  +  a^x  —  ab^  —  b'^x 

CASE  VI. 
75.  To  subtract  one  fractional  quantity  from  another. 

RULE. 

I.  Reduce  the  fraetions  to  a  eommon  denominator. 

II.  Subtract  the  numerator  of  the  subtrahend  from,  the  nu?nrr- 
ator  of  the  minuend,  and  place  the  difference  over  the  common  de- 
nominator. 


54  ELEMENTS    OF    ALGEBRA.  [CHAP.  Ill 

EXAMPLES. 

,.„,  -   ,       .       .  X  —  a  2a  —  4x 

I.  rinu  tne  uiflerence  of  the  Iractions     — -- —    and    - — 

2o  36- 

llere,     (x  —    a)  X  3c  =:  3cx  —  3ac  )     , 

^  >    the  numerators. 

(2a  —  4a;)  X  2b  =  Aab  —  8bx  ) 

And,  2b    x  3c  =  6bc         the  common  denominator. 

3cx  —  3ac        4ab  —  8bx       3cx  —  3ac  —  4ab  +  8bx 


llence, 


6bc  6bc  6bc 


12a;  3a;  30x 

2.  Required  the  difference  of     — -    and    — .  Ans.  -^-. 

3.  Required  the  difference  of  5y   and    -—.  Ans.  — --. 

8  o 

3x  2x  1 3x 

4.  Required  the  difference  of    -—    and    -— .  Ans.  — -— . 

^  7  9  63 

5.  Required  the  difference  of    — r —    and    -r-. 

0  a 

dx  -\-  ad  —  be 

^"^-  Vd — 

3x  A-  a  2a;  +  7 

6.  Required  the  difference  of    — -r —    and    — - — . 

24a;  -\-  Sa  —  lObx  —  3bb 

Ans.  ; . 

406 

7.  Required  the  difference  of    3a;  H — ~    and    x • 

b  c 

ex  -^  bx  —  uh 


be 


Ans.  2x  + 

CASE  vn. 

76.  To  multiply  fractional  quantities  together. 

RULE. 

If  the  quantities  to  be  midtipJied  are  mixed,  reduce  them  fo  a 
fractional  form ;  then  multiply  the  numerators  together  for  a  nu- 
merator and  the  denominators  together  for  a  denominator. 


CHAP.  III.]  ALGEBRAIC    FRACTIONS. 


EXAMPLES. 


55 


bx  c 

i.  Multiply     a -\ by    — . 

bx       a^  +  hx 

First,  -      -       -       a  -\ = ; 

a  a 

d^  -\-hx         c         a^c  +  hex 

Hence,        -        X  -j-  = 9 5 

a  a  ad 

r     3x         ,     3a  .         9ffT 

2.  Required  the  product  of    —    and    — .  Ans.  —j-. 

2x  Sx"^  3a;' 

3.  Required  the  product  of    ^    and    — -.  Avs.  — — . 

2a!;       Sab  ,     3r/r 

4.  Find  the   continued   product  ot     — ,     ,    and    -— -. 

*  a         c  ■lo 


Ans.   dux. 


hx  ft 

5.  It  is  required  to  find  the  product  of     b  -\ and    — . 

oh  4-  bx 

Ans. . 

,r 

a-2  _  ^2  a:2  4-  b"^ 

6.  Required  the  product  of    — ; and    — — ■ . 

^  ^  be  6  +  c 

x^  —  h* 
Ans. 


b'^c  +  hc^ 

a;  -f-  1  X  —  1 

7.  Required  the  product  of    x  -\ ,    and     — T~T' 

ax"^  —  ax  +  a'2  —  1 


Ans. 


+  ah 


.                                                    ax  a''  —  X 

'^  8.  Required  the  product  of     a  -\ by    — — — ^. 

a^{a  +  x) 

Ans.   — r. 

x  (1  -+-  x] 

CASE  VIII. 
77.  To  divide  one  fractional  quantity  by  another. 

RULE. 

Reduce  the  mixed  quantities,  if  there  are  any,  to  a  fractiounl 
firm  :  theji  invert  the  terms  of  the  divisor  and  multiphj  the  frac- 
tions together  as  in  the  last  case. 


56  ELEMENTS    OF    ALGEBRA.  [CHAP    III 


EXAMPLES. 


X..  .,                                     b  f 

1.  Divide  -       -       -       a by    ■^. 

b        2ac  —  b 


a  — 


2c  2c 

Hence,       g- ^  ^  L  ^''j^  yS^'^acg -bg 

2c       g            2c            f  2cf 

-.    T          7a;    ,,..,,,        12  ,         9\x 

2.  Let     —    be  divided  by    — .  Ans.   —  . 

5                            -^     13  60 

Ax^  Ax 

3.  Let     be  divided  by  5x.  Ans.  — '■. 

1                             ^  35 

^-         a+l,,.  .,-,        2x  ^         a?+l 

4.  Let     — - —     be  divided  by     — .  Ans.   

6                               -^3  4,r 

5.  Let be  divided  by     — .  Ans. 


X  -\  '2  X  —  \ 

6.  Let     -—    be  divided  by     — -.  Ans.   . 

3  -^      36  2a 

m     T  X  —  b  l-.ji,  ^cx  ^  x  —  h 

7.  Let     -r— T-    be  divided  by    — -.  Ans.  — - — . 

Scd  ^      Ad  6c2a: 

„    -  a;*  —  6*  ,       ,..,,,        x'^-\-bx 

8.  Let     — — — — —    be  divided  by    r-. 

x^  —  2bx-\-b^  ^      x  —  b 

62 

Ans.  X  -\ . 

X 

„    ^.  .,        ax  —  1    ,  a  ,        aa;(l  +  a;)  —  ac— 1 

9.  Divide     — by -.         Ans.  -^ — - — '- . 

1— a;l  —  x^  a 


10.  Divide by -.  Ans.  —  (1  -j-  o). 


a+  1    ,     •  1  4-  a 

If  we  have  a  fraction  of  the  form 

a 


T^'^' 

a 

-  c    and 

=  c  ;  that  is. 

—  0 

-b~ 

we  may  observe  that 

— T  =  —  c,    also 

0 

The  sign  of  the  quotient  will  be  changed  by  changing  the  sign 
Cither  of  the  numerator  or  denominator,  but  will  not  be  affected  by 
changing  the  signs  of  both   the  terms. 


CHAP.  III.]  ALGEBRAIC    FRACTIONS.  57 

78.  We  will  add  but  two  propositions  more  on  the  subject  of 
fractions. 

If  the  same  number  be  added  to  each  of  the  terms  of  a  proper 
fraction,  the  new  fraction  resulting  from  this  addition  will  be  greater 
than  the  first ;  but  if  it  be  added  to  the  terms  of  an  improper 
fraction,  the  resulting  fraction  will  be  less   than   the  first. 

a 
Let  the  fraction  be  expressed  by    — ,    and  suppose    a  <^b. 

Let   m   represent   the   number  to  be  added  to  each  term :    then 

the  new  fraction  becomes . 

b  -{-  m 

In  order  to  compare  the  two  fractions,  they  must  be  reduced 
to  the  same  denominator,  which  gives  for 


the  first  fraction, 


and  for  the  new  fraction, 


a        ab  -{-  am 
b         b'^  -\-  bm 

a  -\-  m        ab  -\-  bm 


b  -\~  m        b"^  -\-  bm 

Now,  the  denominators  being  the  same,  that  fraction  will  be 
the  greatest  which  has  the  greater  numerator.  But  the  two  nu- 
merators have  a  common  part  ab,  and  the  part  bm  of  the  sec- 
ond is  greater  than  the  part  am  of  the  first,  since    b  ^  a:   hence 

ab  -\r  bm  "^  ab  -{•  am  ; 

that  is,  the   second  fraction  is  greater  than  the  first. 

If  the  given  fraction  is  improper,  that  is,  if  a  >  J,    it  is  plain 

•hat  the  numerator  of  the    second   fraction  will  be    less  than  that 

of  the  first,  since  bm  would  then  be  less  than  am. 

If  the  samjB  number  be  subtracted  from  each  term  of  a  proper 
fraction,  the  value  of  the  fraction  will  be  diminished;  but  if  it  be 
subtracted  from  the  terms  of  an  improper  fraction,  the  value  of  the 
fraction  will  be  increased. 

Let  the  fraction  be   expressed  by    — ,    and  denote  the  number 

to  be  subtracted  by  m. 

Inen,  — z=    the  new  fraction 

0  —  m 


58  ELEMENTS    OF    ALGEBRA. 

By  reducing  to   ilie   same   denominator,  we  have, 
a         ah  —  am 


and 


b         IP'  —  bm 
a  —  m       ah  —  hm 


b  —  m        b"^  —  bm 
Now,  if  we  suppose    a  <^b,    then   am  <  hm;    and   if  am  <C  bm 
then  will 

ab  —  am  '^  ab  —  hm  : 
that  is,  the  new  fraction  will  be  less  than  the  first. 
If  a'^b,    that  is,  if  the  fraction  is  improper,   then 
am  >  bm,    and    ah  —  am  <^  ab  —  hm, 
that  is,  the  new  fraction  will  be  greater  than  the  first. 

GENERAL    EXAMPLES. 

1.  Add to  Ans.  -\ f. 

1  —  a:^  1+0.-2  I  —  X* 

.  , ,  1  1  .  *-^ 

2.  Add     to .  Ans. 


I  +  X  \  ~  X  1  —  x2 

a  -\-  b        .        a  —  b  4ab 

3.  From    7    take r.  Ans. 


a  —  b                a  -{-  b  cP-  —  h"^ 

4.  From take -.  Ans. 


\  —  X-  1  -f  .1-2  1  —  x^ 

,.  ,  .  ,        0-2  _  9a:  +  20     ,        x2  -  ]  3.r  +  42 
^-  ^^^"^^^P^>^  .2_6,  by         ^,_,^       ■ 

x"^  —  WxA  28 


Ans. 


x" 


y^    ^4  Xp'  -\-    JlX 

6.  Multiply      ^    ,    ^. — —Tz    by    —•  ^"•^-  *^  +  ^'^^■ 

x^  -\-  2bx  -{-  h^  X  —  b 

«    T^-   -J  a  +  X       a  —  X  a  -\-  X       a  —  x 

7.  Divide 1 —    by — . 

a  —  a:       a  -\-  x  a  —  x       a  -\-  x 

o2  _|_   r2 

Ans. 

"Zax 

^.   .,  ,        "  —  1.        ,        n  —  \ 

8.  Divide        1  H -— -    by    1 -— -.  Ans.  n. 

n  -\-  1  n  -f  1 


CHAP.  IV.J  EQUATIOXS    OF    THE    FIRST    DEGRFE.  59 


CHAPTER  IV. 

OF    EtiUATlONS    OF    THE    FIRST    DEGREE. 

79.  Ax  Equation  is  the  algebraic  expression  of  two  equai  quan- 
ties   with  the  sign  of  equality  placed  between  them.     Thus, 

is   an   equation,  in  which  x  is   equal  to  the  sum  of  a  and   b. 

80.  By  the  definition,  every  equation  is  composed  of  two  parts, 
separated  from  each  other  by  the  sign  =.  The  part  on  the 
left  of  the  sign,  is  called  the  first  member,  and  the  part  on  the 
right,  is  called  the  second  member;  and  each  member  may  be 
composed  of  one  or  more  terms. 

81.  Every  equation  may  be  regarded  as  the  enunciation,  in  al- 
gebraic language,  of  a  particular  problem.     Thus,  the  equation 

X  -\-  X  z^  30, 

is  the  algebraic  enunciation  of  the  following  problem : 

To  find  a  number  which,  being  added  to  itself,  shall  give  a  sum 
equal  to   30. 

Were  it  required  to  solve  this  problem,  we  should  first  express 
it  in  algebraic   language,  which  would  give  the  equation 

X  -\-  X  =^  30, 
by  adding  x  to  itself,  -  -  2x  =  30, 
and  dividing  by  2,        -       -         a:  ==  15. 

Hence  we  see  that  the  solution  of  a  problem  by  algebra,  con- 
sists of  two  distinct  parts :  viz.,  the  statement,  and  the  solution  oS 
ail  equation 


GO  ELEMENTS    OF    ALGEBRA.  '    fCHAP.  IV. 

The  STATEMENT  consists   in  finding  an  equation  which  shall   ex 
press  the  relation  between  the  known  and  unknown  quantities  of  the 
problem. 

The  SOLUTION  of  the  equation  consists  in  finding  such  a  value 
for  the  unknown  quantity  as  being  substituted  for  it  in  the  equa- 
tion null  satisfy  it ;  that  is,  make  the  first  member  equal  to  the 
second. 

82.  An  equation  is  said  to  be  verified,  when  such  a  value  is 
substituted  for  the  unknown  quantity  as  will  prove  the  two  mem- 
bers of  the  equation  to  be  equal  to  each  other. 

83.  Equations  are  divided  into  classes,  with  reference  to  the 
highest  exponent  with  which  the  unknown  quantity  is  affected. 

An  equation  which  contains  only  the  first  power  of  the  un- 
known quantity,  is  called  an  equation  of  the  first  degree:  and 
generally,  the  degree  of  an  equation  is  determined  by  the  greatest 
of  the  exponents  with  which  the  imknown  quantity  is  affected, 
without  reference  to  other  terms  which  may  contain  the  unknown 
quantity  raised  to  a  less  power.     Thus, 

ax    -\-    b    =:  ex  -\-  d        is  an  equation  of  the  1st  degree. 

2x'^  —  3a;   =  5    —  2x'^     is  an  equation  of  the  2d    degree. 

4x^  —  5x2  =:  3x  +  11       is  an  equation  of  the  3d    degree. 

It  more  than  one  unknown  quantity  enters  into  an  equation,  its 

degree  is  determined  by  the  greatest   sum  of  the    exponents  with 

which  the  unknown  quantities   are   affected   in    any  of  its  terms. 

Thus. 

xy     +  bcx  =  d*     is  of  the  second  degree. 
xyz"^  4-  cx^  =  a^     is  of  the  fourth  degree. 

84.  Equations  are  also  distinguished  as  numerical  equations  and 
literal  equations.  The  first  are  those  which  contain  numbers  only, 
with  the  exception  of  the  unknown  quantity,  which  is  always  de- 
noted by  a  letter.     Thus, 

4«  —  3  =  2a;  +  5,     3a;2  —  a;  =  8, 
are  numerical  equations.     They  are  the  algebraical  translation  of 
problems,  in  which  the  known  quantities  are  particular  numbers. 

A  literal  equation  is  one  in  which  a  part,  or  all  of  the  known 
quantities,  are  represented  by  letters.     Thus, 

bx^  -\-  ax  —  3x  =  5,    and    ex  +  dx^  =  c  -{-f 
are  literal  equations. 


CHAP.  IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  61 

85.  It  frequently  occurs  in  Algebra,  that  the  algebraic  sign  + 
or  — ,  which  is  written,  is  not  the  true  sign  of  the  term  before 
which  it  is  placed.  Thus,  if  it  were  required  to  subtract  —  b 
from  a,  we  should  write 

a  —  (  —  h)  :=  a  -\-  h. 

Here  the  true  sign  of  the  second  term  of  the  binomial  is  plus 
although  its  algebraic  sign,  which  is  written  in  the  first  member 
of  the  equation,  is  — .  This  minus  sign,  operating  upon  the  sign 
of  b,  which  is  also  negative,  produces  a  plus  sign  for  b  in  the 
result.  The  sign  which  results,  after  combining  the  algebraic 
sign  with  the  sign  of  the  quantity,  is  called  the  essential  sign  of 
the  term,  and  is  often  different  from  the  algebraic  sign. 

By  considering  the  nature  of  an  equation,  we  perceive  that  it 
must  possess  the  three  following  properties : 

1st.  The  two  members  are  composed  of  quantities  of  the  same 
kind. 

2d.  The  two  members  are  equal  to  each  other. 

3d.  The  essential  sign  of  the  two  members  must  be  the  same. 

86.  An  axiom  is  a  self-evident  proposition.  We  may  here  stale 
the  following : 

1.  If  equal  quantities  be  added  to  both  members  of  an  equa- 
tion, the  equality  of  the  members  will  not  be   destroyed. 

2  If  equal  quantities  be  subtracted  from  both  members  of  an 
equation,   the  equality  will  not  be   destroyed. 

3.  If  both  members  of  an  equation  be  multiplied  by  the  same 
number,  the  equality  will  not  be  destroyed. 

4.  If  both  members  of  an  equation  be  divided  by  the  same 
number,  the  equality  will  not  be  destroyed. 

Solution  of  Equations  of  the  First  Degree. 

87.  The  transformation  of  an  equation  is  any  operation  by 
which  we  change  the  form  of  the  equation  without  affecting  the 
equality  of  its  members 

First  Transformation. 

88.  When  some  of  the  terms  of  an  equation  are  fractional,  to 
reduce  the  equation  to  one  in  which  the  terms  shall  be  entire. 


62^  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV. 

Take  the  equation, 

2x        3  X 

First,  reduce  all  the  fractions  to  the  same  denominator,  by  the 
Known  rule ;  the  equation  then  becomes 

4aT       5ix       12a; 
"72"~  W^   72""  ■^^' 

If  now,  both  members  of  this  equation  be  multiplied  by  72,  the 
equality  of  the  members  will  be  preserved,  and  the  common  de- 
nominator will  disappear  ;    and  we  shall  have 

48x  —  5ix  +  12a;  z=  792  ; 
or  dividing  by  6,         8x  —    9x -\-    2x  =  132. 

.89.  The  last  equation  could  have  been  found  in  another  man- 
ner by  employing  the  least  common  multiple  of  the  denominators. 

The  common  multiple  of  two  or  more  numbers  is  any  number 
which  each  will  divide  without  a  remainder ;  and  the  least  com- 
mon multiple,  is   the  least  number  which  can  be   so   divided. 

The  least  common  multiple  can  generally  be  found  by  inspec- 
tion. Thus,  24  is  the  least  common  multiple  of  4,  6,  and  8  ;  and 
12  is  the  least  common  multiple   of   3,  4,  and  6. 

Take  the  last  equation, 

2x        3  X 

a;-|----=  11. 

3         4  6 

We  see  that  12  is  the  least  common  multiple  of  the  denomina- 
tors, and  if  we  multiply  each  term  of  the  equation  by  12,  divi- 
ding at  the  same   time  by  the  denominators,  we  obtain 

Qx  —  ^x  +  2x  =  132, 
the  same   equation   as  before   found. 

90.  Hence,  to  transform  an  equation  invohang  fractional  terms 
to  one  involving  only  entire  terms,  we  have  the  following 

RULE. 

Form  the  least  common  mnUiplc  of  all  the  denominators,  and  then 
nnihiphj  every  term  of  the  equation  hy  it,  reducing  at  the  same 
tunc   the   fractional   to   entire  terms. 


CHAP.    [V.]  EQUATIO^f«    OF    THK    FIRST    DEGREE.  63 

EXAMPLES. 

1.  Reduce 1 3  =  20,    to  an  equation  involving  entire 

5         4 

terms. 

We    see,    at  once,    that   the   least  common   multiple    is    20,    by 
uliich  each  term  of  the   equation  is  to  be  multiplied. 

^  r.  20       ^ 

Now,  —  X  20  =  a;  X  -^  =  4a;, 

5  o 

^  ^r.  20  ^ 

and  ,  —  X20=:a;x— -  =  5x; 

4  4 

that  is,   we   reduce   the   fractional   to  entirfj   terms,   by  multiplying 

the   numerator  by   the   quotient   of  the    common    multiple    divided   by 

the  denominator,  and  omitting  the   denominators. 

Hence,  the  transformed  equation  is 

4a;  -f  5.r  -  60  =  400. 

2.  Reduce \-  — 4  —  3     to    an    equation    involving    only 

entire  terms.  Ans.  7a;  +  5a;  —  140  =  105. 

a         c 

3.  Reduce    -— r'\~f=g    to  an  equation  involving  only  en- 

0         d 

tire  terms.  Ans.  ad  —  be  -{-  hdf  ■—  bdg. 

4.  Reduce  the  equation 

ax       2c'^x  4bc^x        5a^        2c^        „, 

7-  +  4a  =  — —-{ 36 

0  ab  a-^  0''  a 

to  one  involving  only  entire   terms. 

Ans.  a^bx  —  2a%c'^x  +  Aa^b'^  =  ib^-'^x  —  5a^  +  202^^  _  2a%^. 

Second  Transformation. 

91.  When  the  two  members   of  an  equation  are   entire  polyno- 
mials to  transpose  certain  terms  from  one  member  to  the  other. 

Take  for  example  the  equation        5x  —  6  =  8  +  2.t. 

If,  in  the  first  place  we  subtract  ") 
2a;  from  both  members,  the  equality  V  5x  —  6  —  2a:  =:  8  -f-  20"  —  2.>-.- 
will  not  be  destroyed,  and  we  have 
or,  by  reducing    the   terms    in    the 
second  member. 


>   5a;  —  6  —  2a;  =  8. 


64  ELEMENTS  OF  ALGEBRA  [CHAP.  IV. 

Whence  we  see  that  the  term  2x,  which  was  additive  in  the 
second  member,  becomes   subtractive  in  the   first. 

In  the  second  place,  if  we  add  6  '\ 
to  both  members,   the   equality  will  >  5a;  —  6  —  2a; +6  =  8  +  6; 
still  exist,   and  we  have  ) 

or,   since   —  6  and    +  6   destroy  each  other      5x  —  2a?  =  8  +  6. 

Hence,    the    term  which  was    subtractive    in   the    first   member, 
passes  into  the  second  member  with  the  sign  plus. 

For  a  second  example,  take  the  equation 

aa?  +  &  =3  fZ  —  ex.  ' 

If  we   add   ex  to   both  ^ 
members    and    subtract   b,  >  ax  +  b  +  cx  — b  =  d  —  cx-\-cx  — b: 
the  equation  becomes  ) 

or  reducing         -       -       -       qx  -\-  ex  =  d  —  b. 

Hence,  we  have  the  following  principle : 

Any  term  of  an   equation  may  be   transposed  from   one   member 
to  the   other  by   changing  its  sign. 

92.  We  will  now  apply  the  preceding  principles  to  the  resolu' 
tion  of  equations. 

1.  Take  the  equation     4a;  —  3  =  2a;  +  5. 

By  transposing  the  terms    —  3  and  2a;,  it  becomes 

4a;  —  2x  =  5  +  3  ; 

and  by  reducing  2a;  =  8 : 

8 
di^ading  by  2  a?  =  -  =  4. 

Now,  if  4  be  substituted  in  the   place  of  x  in   the    given   equa- 
tion, it  becomes 

4x4  —  3=2x4  +  5, 
that  is,  13  =  13. 

Hence,  4  is  the  true  value  of  x ;    for,  being  substituted  for  x  in 
the  given  equation,  that  equation  is  verified. 

2.  For  a  second  example,  take  the   equation 

5x       Ax  7        13a; 

12  "Y"        ~  Y         6~' 


CHAP.   IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  6i> 

By  making  the  denominators   disappear,  we  have 

10^  —  32a:  —  312  =    21  —  52a? 

b)'  transposing       10a;  —  32a;  +  52a:  =    21  +  312 

by  reducing  30a;  =  333 

333       111 
dividing  by  30  x  =  — —  =  — —  =  11.1  ; 

*     -^  30         10  ' 

a   result   which,    being    substituted    for    x,   will   verify   the    givct 
equation. 

3.  For  a  third  example   let  us  take  the  equation 
(3a  —  x)  [a  —  6)  +  2aa;  ■=:  4b  (x  -\-  a). 

It  is  first  necessary  to  perform  thd  multiplications  indicated,  in 
order  to  reduce  the  two  members  to  polynomials,  and  thus  be 
able  to  disengage  the  unknown  quantity  x  from  the  known  quan- 
tities. Having  performed  the  multiplications,  the  equation  be- 
comes, 

3a^  —  ax  —  Sab  -{-  bx  -\-  2ax  =  Abx  +  4a5 ; 
by  transposing      —  ax  -\-  bx  -{■  2ax  —  4bx  =.  4ab  +  Sab  —  Sa^, 
by  reducing  ax  —  35a;  =  7ab  —  Sa^  ; 

or,  (Art.  48),  (a  —  Sb)x  =  lab  —  Sa^. 

Dividing  both  members  by   a  —  Sb,   we  find 

_  lab  —  3a2 
*■"     a  —  Sb' 
93.  Hence,  in  order   to  resolve   any  equation   of  the    first  de- 
^ee,  we  have  the  following  general 

RULE. 

I.  If  the  equation  contains  fractional  terms,  reduce  it  to  one  tn 
which  all  the  terms  shall  be  entire,  and  then  transpose  all  the  terma 
affected  with  the  unknown  quantity  into  the  first  member,  and  all 
the  known  terms  into  the  second. 

H.  Reduce  to  a  single  term  all  the  terms  involving  the  unKnowi> 
quantity :  this  term  will  be  composed  of  two  factors,  one  of  whicJi 
will  be  the  unknown  quantity,  and  the  other  all  its  co-efficients  con- 
nected by  their  respective  signs. 

111.  Then  divide  both  members  of  the  equation  by  the  multiplier 
of  the  unknoion  quantity. 


65  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV. 

EXAMPLES. 

1.  Given    3a;  —  2  +  24  =  31    to  find  x.  Ans.  x  =  3. 

2.  Given    a;+18  =  3a?  —  5    to  find  x.  Ans.  x  =  11|. 

3.  Given    6  -  2a;  +  10  =  20  —  3x  —  2    to  find  x. 

Ans.  X  =  2. 

4.  Given    x-{-— x-{----x=:ll     to  find  x.         Ans.  x  =  6. 

1  fi 

5.  Given    2x —  x  +  1  =  5x  —  2   to  find  a;.     Ans.  x  =  — . 

2  7 


6.  Given    3ax  -\ 3  =  bx  —  a   to  find  x. 


e    —3a 

Ans.   X  =: -. 

6a  —  2b 


-.    /-•          X  -~  3    ,    X        ^„       a?  —  19  ^    , 

7.  Given    — f-  —  =  20 to  find  x. 

.i  o  2 


„^.           a;  +  3,a;         ^       x  —  5  ^, 

8.  Given    — 1-  —  =  4 —    to  find  x 


Ans.  X  =  23^. 

t. 

Ans.  0?  =  3y3. 


„     _,.           ax  —  b        a         bx        bx  —  a  „    . 

9.  Given    — 1-  —  =  — —    to  find  x. 

4  o  ^  o 


Ans.  X  = 


3a  ~2b' 


10.  Given ; 4  =r  f.    to  find  x. 

c  d  *^ 


cdf  +  Acd 

Ans.  X  ■=.  -^ — . 

3aa  —  2bc 


. ,     ^.           Sax  —  b        3b  —  c        ^        ,  ^    . 

1 1 .  Given     — —  =  4  —b,    to  find  x. 


56  -r  9i  -  7c 
Ans.  X  = 


ir.     n-  ^  X  —  2    _      X  13  -     - 

12.  Given    — f-  _  =  _,    to    find  x. 

O  o  id  o 


16a 


Ans.  X  =.  10. 


(3.   Given -\ =  f,    to  find  a:. 

a         b         c         a       -^ 

,  abcdf 

Ans.  X  —  -^ 


ocd  —  acd  -(-  adb  —  abc 


CHAP    TV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  67 

3^ 5       4^ 2 

14.   Given    a — 1 — —  =  x  -\-  1,  to  find  a;. 


Ans.  X  =  6. 


3C         Qx        cc  ^—  3 

15.  Given    —  — —  =:  —  12^,    to  find  re. 

7         9  5  *^ 


^    ^.                      4x  —  2        3x  —  1 
16.  Given    2x r =  — ^ — ,    to  find  x. 


Ans.  «  =  ]  4 


Ans.  a;  =  3- 


17.  Given    Sx  -\ —  —  x  -\-  a,    to  find  x. 

Za  +  d 

18.  Find  the  value  of  x  in  the   equation 

{a  -{-b){x-b)  4ab  —  62  a^  —  bx 

3a  — 1 — 7 2x  -\  - — . 

a  —  0  a  -\-  0  0 

a*  +  3a^  +  4a^^  —  6ab^  +  26* 


Ans.  X  = 


2b  (2a2  +  ai  —  b"^) 


Questions  producing  Equations  of  the  First  Degree,  i?ivolving 
hut  one    Unknown    Quantity. 

94.  It  has  already  been  observed  (Art.  81),  that  the  solution 
of  a  problem  by  Algebra,  consists  of  two  distinct  parts 

1st.  The  statement;    and 

2d.   The   solution  of  the  equation. 

We  have  already  explained  the  methods  of  solving  the  equa- 
tion ;  and  it  only  remains  to  point  out  the  best  manner  of  making 
the  statement. 

Tliis  part  cannot,  like  the  second,  be  subjected  to  any  w^ell- 
defined  rule.  Sometimes  the  enunciation  of  the  problem  furnishes 
the  equation  immediately ;  and  sometimes  it  is  necessary  to  dis- 
cover, from  the  enunciation,  new  conditions  from  which  an  equa- 
tion may  be  formed. 

The  conditions  enunciated  are  called  explicit  conditions,  and 
those  which  are   deduced  from  them,  implicit  conditions. 

In  almost  all  cases,  however,  we  are  enabled  to  discover  the 
equation  by  applying  the  following 


ELEMENTS    OF    ALGEBRA.  [CHAP.  IV. 


RULE. 


Represent  the  unknown  quantity  by  one  of  the  final  letters  of  the 
alphabet,  and  then  indicate,  by  means  of  the  algebraic  signs,  the 
same  operations  on  the  known  and  unknown  quantities,  as  would 
verify  the  value  of  the  unknown  quantity,  were  such  value  known. 

QUESTIONS. 

1.  Find  a  number  such,  that   the   sum   of  one   half,  one   third 
and  one  fourth  of  it,  augmented  by  45,  shall  be  equal  to  448. 
Let  the   required  number  be  denoted  by  -       -       -       -        at . 

Then,  one  half  of  it  will  be  denoted  by  - 


one  third  of  it    -      -      -      -      by 


one  fourth  of  it  -      -      -      -      by  - 

And  by  the  conditions,     — -  +  —  +  — -  +  45=  448, 

Now,  by  subtracting  45  from  both  members, 

^  +  ^  +  ^^403. 
2  ^   3   ^  4 

By  making  the  terms  of  the  equation  entire,  we  obtain 

6a;  +  4x  +  3a;  =  4836  ; 

or        -       -  13x=4836. 

4836 
Hence       -  a;  =  -— -  =  372. 

Let  us   see   if  this  value  will  verify  the  equation  of  the   prob- 
lem.    We  have 

?!^  +  ^  +  !Z!  +  45  =  186  -f-  124  +  93  +  45  =  448. 
2^3^4^ 

2.  What  number   is   that   whose   third  part   exceeds  its  fourth. 

by  16. 

Let  the  required  number  be  represented  by  x.     Then 

—  X  =z     the  third  part. 
3  ^ 

—  X  =z     the  fourth  part. 
4 


CHAP.  IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  69 

And  by  the  question      -—  x x  =    16. 

or,        -       -       -       -       4a;  —  3a;  =  192. 

a;  =  192. 

Veri/ication. 
1|?  _!!_-=  64 -48  =16. 

3.  Out  of  a  cask  of  wine  which  had  leaked  away  a  third  part, 
21  gallons  were  afterward  drawn,  and  the  cask  being  then 
gauged,  appeared  to  be  half  full :    how  much  did  it  hold  ? 

Suppose  the  cask  to  have  held  x  gallons. 

X 

Then,         —  =     what  leaked  away. 

And  "5"  +    21  =    what  leaked  out,  and  what  was  drawn 

X  1 

Hence,       -^-f    21=— a;     by  the  question. 

or  2x  +  126  =:  3a:. 

or  —    X    =  —  126. 

or  a;    =        126, 

by  changing  the  signs  of  both  members,  which  does  not   destroy 
their  equality. 

Verification. 

126  126 

^-  +  21  =  42  +  21  = =  63. 

3  2 

4.  A  fish  was  caught  whose  tail  weighed  9?i. ;  his  head  weighed 
as  much  as  his  tail  and  half  his  body,  and  his  body  weighed 
as  much  as  his  head  and  tail  together :  what  was  the  weight  of 
the  fish? 

Let       -       -         2a;  =     the  weight  of  the  body. 
Then    -       -  9  +  a;  =     weight  of  the  head. 
And  since  the  body  weighed  as  much  as  both  head  and  tail 
2a?  =  9  +    9  +  a; 
or-       -       -       2a;  —  a;  =  18 
and         -       -       -  a;  =  18. 


70  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV 

Verification. 

2x  =z  36  lb  =:     weight  of  the  body. 
9  -\-  X  z=  27  lb  =z     weight  of  the  head. 
9/5  =     weight  of  the  tail. 


Hence,  72/6  =     weight  of  the  fish. 

5.  A  person  engaged  a  workman  for  48  days.  For  each  day 
that  he  labored  he  received  24  cents,  and  for  each  day  that  he 
was  idle,  he  paid  12  cents  for  his  board.  At  the  end  of  the  48 
days,  the  account  was  settled,  when  the  laborer  received  504 
cents.  Required  the  iiumber  of  working  days,  and  the  number  of 
days  he  was  idle. 

If  these  two  numbers  were  known,  by  multiplying  them  respec- 
tively by  24  and  12,  then  subtracting  the  last  product  from  the 
first,  the  result  would  be  504.  Let  us  indicate  these  operations 
by  means  of  algebraic  signs. 

Let     -       -       a;    =     the  number  of  working  days. 

Then      48  —  a;    =     the  number  of  idle  days. 

24  X  a:    =     the  amount  earned,  and 

12(48  —  a?)  =     the  amount  paid  for  his  board. 

Then  24a?   —  12  (48  —  x)     =    504    what  he  received. 

or  24a;  —  576  +  12a;  =    504. 

or  36a;  =  504  -|-  576  =  1080 

1080  ,  ,  .        , 

and  X  =  — ^TT-  =  30     the  working  days. 

3b 

whence,  48  —  30  =  18     the  idle  days. 

Verification. 

Thirty  day's  labor,  at  24  cents  a  day, 
amounts  to 30x24  =  720  cts. 

And  18  days'  board,  at  12  cents  a  day, 
amounts  to 18x12  =  216  cts. 

And  the  amount  received  is  their  difference  504. 


CHAP.  IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  71 

General  Solution. 

the  whole  number  of  working  and  idle  days, 
the  amount  received  for  each  day  he  worked, 
the  amount  paid  for  his  board,  for  each  idle  day. 
the  balance  due,  or  the   result  of  the  account, 
the  number  of  working  days, 
the  number  of  idle  days, 
what  he  earned ; 
the  amount  deducted  for  board. 
The  equation  of  the  problem  will  then  be, 
ax  —  i  («  —  x)  =z  c 
whence  ax  —    bn    -\-  bx  z=.  c 

[a  -\-  b)  X  =:  c   -\-  bn 
c   -\-  bn 


^     ,    ^..       an  -\-  bn  —  c  —  bn 
and  consequently,  n  —  a;  =  n  — 


L,e1 

t, 

n 

= 

a 

=z 

b 

= 

c 

= 

X 

= 

n 

—  X 

=: 

Then 

> 

ax 

z=: 

and, 

b{n- 

-X) 

= 

or 


a 

+  b 

c 

+  bn 

a 

+  b 

an 

\  —  c 

+  b 


u-\-b 


6.  A  fox,  pursued  by  a  greyhound,  has  a  start  of  60  leaps. 
He  makes  9  leaps  while  the  greyhound  makes  but  6 ;  but  '6 
leaps  of  the  greyhound  are  equivalent  to  7  of  the  fox.  How 
many  leaps  must  the  greyhound  make  to  overtake  the   fox  ? 

From  the  enunciation,  it  is  evident  that  the  distance  to  be 
passed  over  by  the  greyhound,  is  equal  to  the  60  leaps  of  the 
fox,  plus  the  distance  which  the  fox  runs  after  the  gi'eyhound 
starts  in  pursuit. 

Let  X  =  the  number  of  leaps  made  by  the  greyhound  from 
the  time  of  starting  till  he   overtakes   the  fox. 

Now,  since  the  fox  makes  9  leaps  while  the  greyhound  makes 

3 
6,   the    fox  will    make    1  ^,    or     —     leaps    while    the    greyhound 

makes  1  ;  and,  therefore,  while  the  greyhound  makes  x  leaps,  the 

3 
{oji  will  make    —  x    leaps.     Hence, 

3 

60  +  ^*  = 


72  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV 

the  number  of  leaps  made  by  the  fox,  in  passing  over  the  entire 
distance. 

It  might,  at  first,  be  supposed  that  the  equation  of  the  problem 
would  be  obtained  by  placing  this  number  equal  to  x;  but  in 
doing  so,  a  manifest  error  would  be  committed ;  for  the  leaps  of 
lie  greyhound  arps  greater  than  those  of  the  fox,  and  we  should 
ihus  equate  nmnhers  referred  to  different  units.  Hence,  it  is  ne- 
cessary to  express  the  leaps  of  the  fox  by  means  of  those  of 
the  greyhoimd,   or  reciprocally. 

Now,   according  to  the    enunciation,   3   leaps   of  the   greyhound 

are  eq-iivalent  to   7  leaps  of  the   fox  ;    and  hence,   1   leap   of  the 

7 
greyhound     s    equivalent   to    —    leaps  of  the    fox ;    consequently, 

7x 
X  leaps  of  the  gre)  ^'•ound  are  equivalent  to    —    of  the  fox :    that 

is,  had  the  leaps  of  the  grt\^"nund  been  no  longer  than   fhose  of 

the   fox,  he  would  have   made    —    le^.^.?  instead  of  x  leaps. 

7x                    3 
Hence  the  true  equation  is,      — ,=z    60 -| x  ; 

or,  by  making  the  terms  entire    14a;  =  360  +  9a;, 
whence        -     '  -       -       -       -        5a;  =:  360    and    x  =  72. 

Therefore,  the   greyhound  will  make  72  leaps  to  overtake  the  fox, 

3 
and  during  this  time  the  fox  will  make    72  x  —  =108. 

Verification. 

The  72  leaps  of  the  greyhound  are  equivalent  to 

72  X  7 

■ —  =  168    leaps  of  the   fox   =  the  whole  distance. 

And  60  +  108  =  168,  the  leaps  which  the  fox  made  from  the 
beginning. 

7.  A   can  do  a   piece   of  work  alone  in   10  days,  and  B  in  ^3 
days :    in  what  time   can  they  do  it  if  they  work  together  ? 
Denote  the   time  by  x,  and  the  work  to  be  done  by  1.     Then 

m   I   day  A  could   do    —    of  the  work,  and  B   could  do    —    of 


';HAP.  IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  73 


X 


it :    and  in   x   days   A   could  do     —    of   the    work,    and    B 

•^  10  13 

hence,  by  the  conditions  of  the  question, 

-  +  --1 
10^  13 -' 

wliich  gives  13a;  +  lOx  =  130: 

130 
bence,  23a;  =  130,    x  =  — ;—  =  5^|    days. 

8.  Divide  $1000  between  A,  B,  and  C,  so  that  A  shall  have 
$72  more  than  B,  and  C  $100  more  than  A. 

Ans.  A's  share   =  $324,    B's   =  $252,    C's   =  $424. 

9.  A  and  B  play  together  at  cards.  A  sits  down  with  $84 
and  B  with  $48.  Each  loses  and  wins  in  turn,  when  it  ap- 
pears that  A  has  five  times  as  much  as  B.  How  much  did  A 
win?  Ans.  $26. 

10.  A  person  dying  leaves  half  of  his  property  to  his  wife,  one 
sixth  to  each  of  two  daughters,  one  twelfth  to  a  servant,  and  the 
remaining  $600  to  the  poor :  what  was  the  amount  of  his  prop- 
erty ?  Ans.  $7200-. 

11.  A  father  leaves  his  property,  amounting  to  $2520,  to  four 
sons,  A,  B,  C,  and  D.  C  is  to  have  $360,  B  as  much  as  C 
and  D  together,  and  A  twice  as  much  as  B  less  $1000:  how 
much  does  A,  B,  and  D,  receive  ? 

Ans.  A  $760,    B  $880,  D  $520. 

12.  An  estate  of  $7500  is  to  be  divided  between  a  widow,  two 
sons,  and  three  daughters,  so  that  each  son  shall  receive  twice  as 
much  as  each  daughter,  and  the  widow  herself  $500  more  than 
all  the  children :  what  was  her  share,  and  what  the  shai-e  of 
each  child?  r  Widow's  share  $4000. 

Ans.    )  Each  son  $1000. 

(  Each  daughter  $500. 

13.  A  company  of  180  persons  consists  of  men,  women,  and 
children.  The  men  are  8  more  in  number  than  the  women,  and 
the  children  20  more  than  the  men  and  women  together :  how 
many  oi  each  sort  in  the  company  ? 

Ans.  44  men,    36  women,    100  children. 


74  ELEMENTS    OF    ALGEBRA.  [CHAP.   IV 

14.  A  father  divides  $2000  among  five  sons,  so  that  each  elder 
should  receive  $40  more  than  his  next  younger  brother :  what  is 
the  share  of  the  youngest  ?  Ans.  $320. 

15.  A  purse  of  $2850  is  to  be  divided  among  three  persons, 
A,  B,  and  C;  A's  share  is  to  be  to  B's  as  6  to  11,  and  C  is 
to  have  $300  more  than  A  and  B  together :  what  is  each  oiie'.s 
share?  Ans.  A's  $450,    B's  $825,    C's  $1575. 

16.  Two  pedestrians  start  from  the  same  point;  the  first  steps 
twice  as  far  as  the  second,  but  the  second  makes  5  steps  while 
the  first  makes  but  one.  At  the  end  of  a  certain  time  they  are 
300  feet  apart.  Now,  allowing  each  of  the  longer  paces  to  be  3 
feet,  how  far  will  each  have  travelled  ? 

Ans.   1st,  200  feet;    2d,  500. 

17.  Two  carpenters,  24  journeymen,  and  8  apprentices,  re- 
ceived  at  the  end  of  a  certain  time  $144.  The  carpenters 
received  $1  per  day,  each  jour.ieyman  half  a  dollar,  and  each 
apprentice  25  cents :    how  many  days  were  they  employed  ? 

Ans.  9  days. 

18.  A  capitalist  receives  a  yearly  income  of  $2940:  four  fifths 
of  his  money  bears  an  interest  o*"  4  per  cent.,  and  the  remainder 
of  5  per  cent.:  how  much  has  he  at  interest?       Ans.  $70000. 

19.  A  cistern  containing  60  gallons  of  water  has  three  unequal 
cocks  for  discharging  i'  ,  the  largest  will  empty  it  in  one  hour, 
the  second  in  two  hoiirs,  and  the  third  in  three  :  in  what  time 
will  the  cistern  be  emptied   }f  they   all  run  together  ? 

Ans.  32^   min. 

20.  In  a  fi^rtam  orchard  ^  are  apple-trees,  \  peach-trees, 
\  plum-treos,  .  20  cherry-trees,  and  80  pear-trees :  how  many 
trees  in  the  orchard^  A  as.  2400. 

21.  A  farmer  being  an  kid  how  many  sheep  he  had,  answen;d 
that  he  had  them  in  five  fields;  in  the  1st  he  had  \,  in  the 
2d  1,  in  the  3d  i,  in  the  4th  ^^,  and  in  the  5th  450:  how 
many  had  he  ?  Ans.  1200. 

22.  My  horse  and  saddle  together  are  wor'h  $132,  and  tlie 
horse  is  worth  ten  times  as  much  as  the  saddle :  what  is  the 
value  of  the  horse?  Ans.  $120. 


CHAP.   IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  75 

23.  The  rent  of  an  estate  is  this  year  8  per  cent,  greater  than 
it  was  last.     This   year  it  is   $1890:    what  was  it  last  year? 

Ans.  $1750. 

24.  What  number  is  that  from  which,  if  5  be  subtracted,  f  of 
the  remainder  will  be  40  ?  Ans.  65. 

25.  A  post  is  1  in  the  mud,  ^  in  the  water,  and  ten  feet  above 
the  water  :    what  is  the  whole   length  of  the  post  1 

Ans.  24  feet. 

26.  After  paying  I  and  i  of  my  money,  I  had  66  guineas  left 
in  my  purse  :    how  many  guineas   were  in  it  at  first  ? 

Ans.  120. 

27.  A  person  was  desirous  of  giving  3  pence  apiece  to  some 
beggars,  but  found  he  had  not  money  enough  in  his  pocket  by  8 
pence  ;  he  therefore  gave  them  each  two  pence  and  had  3  pence 
remaining:  required  the  number  of  beggars.  Ans.   11. 

28.  A  person  in  play  lost  A  of  his  money,  and  then  won  3 
shillings  ;  after  which  he  lost  i  of  what  he  then  had  ;  and  this 
done,  found  that  he  had  but  12  shillings  remaining :  what  had 
he   at  first  ?  Ans.  20s. 

29.  Two  persons,  A  and  B,  lay  out  equal  sums  of  money  in 
trade;  A  gains  $126,  and  B  loses  $87,  and  A's  money  is  now 
double  of  B's  :    what  did  each  lay  out  ?  Ans...  $300. 

30.  A  farmer  bought  a  basket  of  eggs,  and  offered  them  at  7 
cents  a  dozen.  But  before  he  sold  any,  5  dozen  were  broken 
by  a  careless  boy,  for  which  he  was  paid.  He  then  sold  the  re- 
mainder at  8  cents  a  dozen,  and  received  as  much  as  he  would 
have  got  for  the  whole  at  the  first  price.  How  many  eggs  had 
he  in  his  basket  ?  Ans.  40  dozen. 

31.  A  person  goes  to  a  tavern  with  a  certain  sum  of  money  in 
his  pocket,  where  he  spends  2  shillings ;  he  then  borrows  as 
much  money  as  he  had  left  and  going  to  another  tavern,  he 
there  spends  2  shillings  also ;  then  borrowing  again  as  much 
money  as  was  left,  he  went  to  a  third  tavern,  where  likewise 
he  spent  2  shillings  and  borrowed  as  much  as  he  had  left ;  and 
again  spending  2  shillings  at  a  fourth  tavern,  he  then  had  nothing 
remaining.     What  had  he   at  first  ? 

Ans.  3s.  9d 


76  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV. 

Of  Equations  of  the  First  Degree,   involving  two  or  more 
Unknown    Quantities. 

95.  Although  several  of  the  previous  questions  contained  in  their 
enunciation  more  than  one  unknown  quantity,  we  have  neverthe- 
less resolved  them  all  by  employing  but  one  symbol.  The  rea- 
son of  this  is,  that  we  have  been  able,  from  the  conditions  of  the 
enunciation,  to  represent  the  other  unknown  quantities  by  means 
of  this  symbol  and  known  quantities  ;  but  this  cannot  be  done 
in  all  problems  containing  more  than  one  unknown  quantity. 

To  explain  the  methods  of  resolving  problems  of  this  kind,  let 
us  take  some  of  those  which  have  been  resolved  by  means  of 
one  unknown  quantity. 

1.  Given  the  sum  of  two  numbers  equal  to  a,  and  their  differ 
ence   equal  to  h ;    it  is   required  to  find  the  numbers. 

Let    X  =    the  greater,  and  y  the  less  number 
Then  by  the  conditions  x  -\-  y  =  a  ; 

and  a;  —  y  =:  b. 

By  adding  (Art.  86,  Ax.  1),  2x  —  a  -{-  b. 

By  subtracting  ( Irt.  86,  Ax.  2),    2y  =  a  —  b. 

Each  of  these  equations  contains  but  one  unknown  quantity 

From  the  first  we  obtain  x 

And  from  the  second  y 


2 
a-h 


Verification, 
a  -\-  b       a  —  b       2a  a  -\-  b       a  —  b        2b 

~~2  2~  ~  T  ~  ^ '    ^^       ~2  2~  ~  '2~    ' 

2.  A  person  engaged  a  workman  a  number  of  days,  denoted 
by  n.  For  each  day  that  he  labored  he  was  to  receive  a  cents, 
and  for  each  day  that  he  was  idle  he  was  to  pay  b  cents  for  his 
board.  At  the  end  of  the  n  days,  the  account  was  settled,  when 
the  laborer  received  c  cents.  Required  the  number  of  working 
days  and  the  number  of  days  he  was  idle. 
Let  X  =    the  number  of  working  days. 

y  =    the  number  of  idle  days. 
Then,  ax  =    what  he  earned, 

and  by  —     what  h-e  paid  for  his  board  ; 


CHAP.  IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  77 


and  by  the  question,  we  have       < 


X  -\-    y  =  n 
ax  —  by  =  c. 

It  has  already  been  shown  that  the  two  members  of  an  equa- 
tion can  he  multiplied  by  the  same  number,  without  destroying 
the  equality ;  therefore,  multiply  both  members  of  the  first  equa- 
tion by  b,  the  co-efiicient  of  y  in  the  second,  and  we  have 

the   equation  -       -       -       -       -       bx  -j-  by  =  bn, 
which,  added  to  the  second     -       ax  —  by  =  e, 
gives        -       -       -       -       -       -       ax  -\-  bx  =  bn  -{-  c. 

Whence  ------        x  = ■ — - 

a  -{-  0 

In  like  manner,  multiplying  the  two  members  of  the  first  equa- 
tion by  a,  the  co-efficient  of  x  in  the  second,  it  becomes 

ax  -{-  ay  =  an; 
from  which,  subtract  the  second  equation,     ax  —  by  =  c, 
and  we  obtain    -       -       -       -       -       -       -     ay  -\-  by  =^  an  —  c. 

Whence       --------  «  = -. 

^  a  +  6 

By  introducing  a  symbol  to  represent  each  of  the  unknown 
quantities  of  the  problem,  the  above  solution  has  the  advantage 
of  making  known  the  two  required  numbers,  independently  of  each 
other. 

What  will  be  the  numerical  values  of  x  and  y,  if  we  suppose 

n  =  48,     a  =  24,     b  =  12,    and    c  =  504. 

Elimination. 

96.  The  method  which  has  just  been  explained,  of  combining 
two  equations,  involving  two  unknown  quantities,  and  deducing 
therefrom  a  single  equation  involving  but  one,  may  be  extended 
to  three,  four,  or  any  number  of  equations,  and  is  called  Elimtnc^ 
tion. 

There  are  three  principal  methods  of  elimination : 

1st.  By  addition  and  subtraction. 

2d.    By  substitution. 

3d.    By  comparison. 

We  shall  discuss  these  methods  separately. 


78  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV 

Elimination  hy  Addition  and  Subtraction. 

97.  Before   considering  the    case  of  Elimination,  we   will   ex 
plain  a  new  notation  which  is  about  to  be  used. 

It  often  happens,  in  Algebra,  that  some  of  the  known  quantities 
of  an  equation  or  problem,  though  entirely  independent  of  each 
other  in  regard  to  their  values,  have,  nevertheless,  certain  rela- 
tions which  it  is  desirable  to  preserve  in  the  discussion.  In  such 
case,  the  second  quantity  is  represented  by  the  same  letter,  with 
a  small  mark  over  it.  Thus,  if  the  first  quantity  was  denoted  by 
a,  the  second  would  be  denoted  by  a',  and  is  read,  a  prime 
If  there  were  a  third,  it  would  be  denoted  by  a'^,  and  read, 
a  second,  &c. 

Let  us  now  take  the  two  equations, 
ax  -\-  by  =  c 
a'x  -\-  h'y  =  </. 

If  the  co-efRcients  of  either  of  the  unknown  quantities  were  the 
same  in  both  equations  ;  that  is,  if  a  were  equal  to  a^,  or  b 
to  y,  we  might  by  a  simple  subtraction  form  a  new  equation  that 
would  contain  but  one  unknown  quantity ;  and  from  this  equa- 
tion, the  value  of  that  unknown  quantity  could  be  deduced. 

If,  now,  both  members  of  the  first  equation  be  multiplied  by  i', 
the  co-efficient  of  y  in  the  second,  and  the  two  members  of  the 
second  by  b,  the  co-efficient  of  y  in  the  first,  we  shall  obtain 

ab'x  +  bby  =  b^c 

a'bx  +  bb'y  =  be/ ; 

and  by  subtracting  the  second  from  the  first, 

{ab'  —  a'b)  x  =  b'c  -  b</ ; 

b'c  -  h(/ 

whence,  a?  =  — -. 

ab  —  ab 

If  we  multiply  the  first  of  the  given  equations  by  a',  and  the 
second  by  a,  we  shall  have 

aa'x  -f-  (t'by  =  a'c 

aa'x  +  (ib'y  =  a<^ ; 

and  by  subtracting  the  second  from  the  first. 


CHAP.   IV.J  EQUATIONS    OF    THE    FIRST    DEGREE.  79 

((/J  —  ab')  y  =  a^c  —  a(/  ; 

.  a'c  —  acf 

whence,  y  =  ITT T>\ 

ab  —  ab 

or,  if  we  wish  the  value  for  y  to  have  the  same  denominator 
with  that  for  x,  we  change  the  signs  of  the  numerator  and  de- 
nominator, and  write 

ac'  —  a'c 
y^  ab'  -  a'b  ^ 

The  method  of  elimination  just  explained,  is  called  the  methoa 
by  addition  and  subtraction,  because  the  unknown  quantities  dis- 
appear by  additions  and  subtractions,  after  having  prepared  the 
equations  in  such  a  manner  that  the  same  unknown  quantity  shall 
have  the  same  co-efficient  in  both  equations. 

b'c  —  be'  ac'  —  afc 


The  formulas         x  =  — — ■,    y  = 


ab'-a'V    "       ab'-a'b 
deduced  from  the  equations 

ax  -\-  by  =  c 
a'x  +  b'y  =  d 

will  enable  us  to  write  the  values  of  x  and  y  immediately,  with- 
out the  trouble  of  elimination.  They  contain  the  germe  of  a  gen- 
eral rule,  not  before  given,  for  the  solution  of  all  similar  equations. 

RULE. 

I.  The  first  term  in  the  numerator  for  the  value  of  x,  is  foimd 
by  beginning  at  h'  and  crossing  up  to  c — giving  b'c  ;  the  second 
term  is  found  by  crossing  from  b   to  c' — giving  he'. 

II.  For  the  first  term  in  the  numerator  of  the  value  for  y,  begin 
at  a  and  cross  down  to  c' — giving  ac' ;  and  for  the  second  term, 
cross  from  df  to  c — giving  a'c. 

III.  The  first  term  of  the  common  denominator  is  found  by  cross- 
ing from  a  to  h' — giving  ab' ;  and  the  second,  by  crossing  from 
a'  to  b — giving  a'b. 

The  manner  of  obtaining  these  formulas  will  be  easily  remem- 
bered,  and   their  applications  will  be  found  very  simple. 
1.  What   are  the  values  of  x  and  y  in  the  equations, 

5x  +  7y  =  43 
Ua;  -f  %  =  69. 


80  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV 


=  3 


We  write  immediately, 

_  9  X  43  —    7  X  69  _  387  —  483  _  —    96 
*  ~  5  X     9-  11  X     7  ^    45-77    ~  -    32 

_  5  X  69  -  11  X  43  _  345  -  473  _  -  128  _ 
y  ~  —  32  ~       —  32       "~  —    32  "" 

2.  What  are  the  vakies  of  x  and  y  in  the  equations, 

3a;  -  4-  =  14 
4 

X  —  4y  =  —  11. 

We  write 

_  -4  X  14-(-l  X  -11)  _  -56-V  _  ^ 

*~        3x-4-lX-i       ~-12+i~ 

3  X  —  11  —1  X  14        —33-14       ^ 

y  =. r= =:  4. 

^  -12  +  i  -12+^ 

Elimination  by    Substitution. 
98.  Let  us  take  the  two  equations 

5a;+7y  =  43    and    lla;+9y=69. 
Find  the  value  of  a;  in  the  first  equation,  which  gives 

43  -  7y 

X  = 

5 

Substitute  this  value  of  x  in  the  second  equation,  and  we  have 

43 7y 

11  X  -— ^  +  9y  =  69. 

5 

or  473  —  77y  +  45y  =  345  : 

or  —  32y  =  —  128. 

Hence  y  =  4. 

43  -  28       „ 

And  X  = =  3. 

5 

This  method,  called  the  method  by  substitution,  consists  in  find- 
ing in  one  equation  the  value  of  one  of  the  unknown  quantities, 
as  if  the  others  were  already  determined,  and  then  substituting 
this  value  in  the  other  equations.  In  this  way,  new  equations 
are  formed  from  which  one  of  the  unlmown  quantities  has  been 
eliminated.  We  then  operate  in  a  similar  manner,  on  the  new 
equations. 


CHAP.   IV.]  EQUATIONfS    OF    THE    FIItST    DF.GREE.  SI 

Elimination   htj    Comjiarison. 
99.  Let  us  take  the  two   equations, 

5a;  +  7y  =  43     and     11a:  +  9y  =  69. 
Finding  the  value  of  x  in  the  first  equation,   we  have 

43  —  7v/ 

X  — 

5 

And  finding;  the  value  of  x  in  the  second,  we  obtain 

69  — 9y 

*  =  -n-- 

Let  these  two  values  of   a;  be  placed  equal  to  each  other,   and 

43  —    7y       69  —    9y 

we  have, = r^ ; 

o  11 

or,  473  —  11  y  !^  345  —  45y ; 

or,  -  32y  =  -  128. 

Hence,  y  =    4 

69  —  36 
and,  X  - 


11 

This  method  of  elimination  is  called  the  method  by  compari- 
son, and  consists  in  finding  the  value  of  the  same  unlinown 
quantity  in  all  the  equations,  and  then  placing  those  values  equal 
to  each  other,  two  and  two.  This  will  give  rise  to  a  new  set  of 
equations  containing  one  less  unknown  quantity,  and  upon  whir'i 
we  operate  as  on  the   given   equations. 

The  new  equations  v/hich  arise,  in  the  last  two  methods  of 
elimination,  contain  fractional  terms.  This  inconvenience  is  avoid- 
ed in  the  first  method.  The  metJtod  hy  substitution  is,  however, 
advantageously  employed  whenever  the  co-eftieient  of  either  of 
the  unknown  quantities  in  one  of  the  equations  is  equal  to  unity, 
because  then  the  inconvenience  of  which  we  have  just  spoken 
does  not  occur.  We  shall  sometimes  have  occasion  to  employ 
this  method,  but  generally  the  method  by  addition  and  subtraction 
is  preferable.  When  the  co-efficients  are  nut  too  great,  we  c-<u\ 
perform  the  addition  or  subtraction  at  the  same  time  with  the 
multiplication  which  is  made  to  render  the  co-efficients  of  the 
same  unknown  quantity  equal  to  each  other. 

6 


82  ELEMENTS  OF  ALGEBRA  [CHAP.  IV. 

lUO.  Let  us  now  consider  the  case  of  three  equations  involving 
three   unknown  quantities. 

f  5x  —  6y  +  4^  =  15. 

Take  the  equations,        <  Tac  +  4y  —  3^;  =  19. 
(  2a  +     y  +  6z  =z  46. 

To  eliminate  z  from  the  first  two  equations,  multiply  the  first 
!■  iu.i'ion  by  3  and  the  second  by  4 ;  and  since  the  co-efficients 
oi'  2  have  contrary  signs,  add  the  two  results  together:  this  gives 
a  new  equation         .         .         .         -         -         43a;  —  2y  =  121 

Multiplying  the  second  equation  by  2,  a  fac- 
tor of  the  co-efficient  of  z  in  the  third  equa- 
tion, and  adding  them  together,  we  have  16a?  +  9y  =    84^ 

The  question  is  then  reduced  to  finding  the  values  of  x  and  y, 
which  will  satisfy  these   new  equations. 

Now,  if  the  first  be  multiplied  by  9,  the  second  by  2,  and  the 
results  be  added  together,   we   find 

419x  —  1257,    whence    x  =  3. 

By  means  of  the  two  equations  involving  x  and  y,  we  may  de- 
termine y  as  we  have  determined  x  ;  but  the  value  of  y  may  be 
determined  more  simply,  by  observing,  that  by  substituting  for  x 
its  value   found   above,  the   last  of  the   two  equations   becomes, 

84  —  48 
48  +  9y  =  84,       whence     y  = =  4. 

In  the  same  manner,  by  substituting  the  values  of  x  and  y,  the 

first  of  the  three  proposed  equations  becomes, 

24 
15  —  24  +  4^  =  15,     whence     z  =z  —  =  6. 

4 

101.  Hence,  if  there   are  m  equations   involving   a  like   number 

of  unknown  quantities,  the  unknown  quantities  may  be  eliminated 

by  the  following 

RULE. 

I.  To  eliminate  one  of  the  unknown  quantities,  combine  any  one 
i)f  the  equations  loith  each  of  the  m  —  1  others ;  there  will  thus 
he  obtained  m  —  1  new  equations  containing  m  —  1  unknoion  quan- 
tities. 

II.  Eliminate  another  unknown  quantity  by  combining  one  of  these 
new  equations  with  the  m  —  2  others;  this  toil  give  m  —  2  equa 
tions  containing    m  —  2    unknown  quantities. 


CHAP.   IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  83 

III.  Continue  this  scries  of  operations  until  a  single  equation  is 
obtained  containing  but  one  unknown  quantity,  the  value  of  which 
can  then  he  found.  Then  by  going  back  through  the  series  of  equa- 
tions the  values  of  the  other  unknown  quantities  may  be  successively 
determined. 

102.  It  often  happens  that  some  of  the  proposed  equations  do 
not  contain  all  the  unknown  quantities.  In  this  case,  with  a 
little  address,  the  elimination  is   very  quickly  performed. 

Take  the   four   equations   involving  four  unknown  quantities, 


2a;  —  3y  +  2^  =  13  ^       -     -     (1)     4y  +  2^  =  14     -     -     (3). 
4m  —  2a;  =  30  3       -     -     (2)     5y  +  3m  =  32     -     -     (4). 


By  examining  these  equations,  we  see  that  the  elimination  of 
z  in  equations  (1)  and  (3),  will  give  an  equation  involving  x 
and  y ;  and  if  we  eliminate  m  in  the  equations  (2)  and  (4),  we 
shall  obtain  a  second  equation,  involving  x  and  y.  In  the  first 
place,  the  elimination  of  2,  in  (1)  and  (3)  gives  ly  —  2a;  =  1 
that  of  M,  in  (2)   and   (4),  gives      -         .         -    20y  +  6a;  =  38 

Multiplying   the  first  of  these  equations  by 
3,  and  and  adding,  we  have  -         -         -         -  41y  =  41 

whence       -------  y  =:     \ 

Substituting  this  value  in    7y  —  2a;  =i  1 ,    we  |> 

find .x  = 

Substituting  for  x  its  value   in  equation  (2), 
it  becomes,    4m  —  6  ^  30,    whence         -         -  m  = 

And  substituting  for  y  its  value  in  equation 
(3),  there  results    ------  z  ■=. 

Of 'indeterminate  Problems. 

103.  In    all   the    preceding   reasoning,    we    have    supposed    the 
number  of  equations   equal  to  the  number  of  unknown  quantities 
This  must  be  the  case  in  every  problem,  in  order  that  it  may  be 
determinate ;   that  is,  in  order  that  it  may  admit  of  a  finite  num- 
ber of  solutions. 

Let  it  be  required,  for  example,  to  find  two  quantities  such, 
that  five  times  one  of  them,  diminished  by  three  times  the  other, 
shall  be  equal  to  12. 


84  ELEMENTS    OF    ALOEBRA.  [CHAP.  IV. 

If  we  denote  the  quantities  sought  by  x  and  y,  we  shall  have 
the  equation 

5a;  —  3y  =  12, 

V  .  12  +  3y 

whence,  x  = — -. 

5 

Now,  by  making  successively, 

y  =:  1,     2,       3,       4,       5,      6,  &c., 

o      18      21      24      27      ^     „ 
there  results,     rr  =  3,     — ,     — ,     — ,     —      6,   &c., 
'  '      5         5'      5         5 

and  any  two  corresponding  values  of  x,  y,  being  substituted  in  the 
given  equation, 

5a;  —  3y  =  12 

will  satisfy  it  equally  well :  hence,  there  are  an  infinite  number 
of  values  for  x  and  y  which  will  satisfy  the  equation,  and  conse- 
quently, the  problem  is  indr terminate  ;  that  is,  it  admits  of  an  in- 
finite number  of  solutions. 

If,  however,  we  impose  a  second  condition,  as  for  example, 
that  the  sum  of  the  two  (juaiitities  sliall  be  equal  to  4,  we  shall 
have   a  second  equation, 

a;  4-  y  =  4  ; 

and  this,  combined  with  the  equation  already  considered,  will 
give  determinate  values  for  x  and  y. 

If  we  have  two  equations,  involving  three  unknown  quantities, 
we  can  eliminate  one  of  the  unknown  quantities,  and  thus  ob- 
tain an  equation  containing  two  unknown  quantities.  This  equa- 
tion, like  the  preceding,  woidd  be  satisfied  by  an  infinite  num- 
ber of  values,  attributed  in  succession,  to  the  unknown  quanti- 
ties. Since  each  equation  expresses  one  condition  of  a  problem, 
therefore,  in  order  that  a  prohlcm  may  be  detenninate,  its  enun- 
nation  must  contain  at  least  as  many  different  conditions  as  there 
are  iinknotim  quantities ,  and  these  conditions  must  be  such,  that  each 
of  them  may  be  expressed  by  an  independent  equation  ;  that  ts,  art 
equation  not  produced  by  any  combination  of  the  others  of  the  system. 

If,  on  the  contrary,  the  number  of  independent  equations  ex- 
ceeds the  number  of  utiknown  quantities  involved  in  them,  the 
conditions  which  they  express  cannot  be  fulfilled. 


CHAl'.   IV. J 


EQUATIONS    OF    THE    FIRST    DEGREE. 


86 


For  example,  let  it  be  required  to   find  two  numbers  such  thai 
their  sum  shall  be  100,  their  difference  80,  and  their  product  700. 
The   equations   expressing  these  conditions   are, 

a:  +  y  =  100 

X  —  y  =    80 

and  X  X  y  =  700. 

Now,  the  first  two  equations  determine  the  values  of  x  and  y, 
viz., 

X  =  Q0    and    y  =z  10. 

The    product   of  the    two  numbers  is  therefore  known,  and   equal 
to  900.     Hence,  the  third   condition  cannot  be   fulfilled. 

Had  the  product  been  placed  equal  to  900,  all  the  conditions 
would  have  been  satisfied,  in  which  case,  however,  the  third  would 
not  have  been  an  independent  equation,  since  the  condition  ex- 
pressed by  it,  is  implied  in  the  other  two. 

EXAMPLES. 


1.  Given     2jr +  3^=16,    and    3x— 2y=:ll     to  find  the  values 
of  X  and  y.  Ans.  a;  =  5,    y  =  2. 

2x       37/        9  ,     Sx       2y        61  ^    ,      , 

2.  Given — ^  =  — ,    and — ~  z=  — -      to     find     the 

5         4        20  4         5        120 


values  of  x  and  y. 


A  1  1 

Ans.  X  =.  — ,    y  =^  — . 
2'    -^         3 


3.   Given     -— +  7y  =  99,    and    -^  +  7x  =  51    to  find  the  values 


of  T  and  y. 


A71S.  X  =  7,    y 


14. 


^.  X  V  ,   X  -{-  y       X  2//  —  X 

4.  Given 12=^1- +  8,  and    -— -i  H 8  = -^ h  27 

2  4  5  3  4 

to  find  the  values  of  x  and  y.  Ans.  x  =  60,    y  =  40. 

X  -f       y  +        z  =  29 
X  -{-     2y  +      3^  =r  62 


ft.  Given     < 


1  1  1 


>     to  find  X,  y,  and  2 


Ans.  a;  =  8,    y  =:  9,    ^  =  12 


m 


ELEMENTS    OF    ALGEBRA. 


[CHAP.   IV, 


2a;  +      4y  —      3z  =  22  1 
6    Given     -^      4a;  —     2y  +      5z  —  18   ^     to  find  x,  y,  and  z 
_     6a;  +      7y  —        ;?  :^  63  J 

Aw5.  a;  =  3,    y  =  7,    ^  =  4. 


7.  Given     -: 


111,. 

-3-^  +  T^  +  T"  =  '' 

1  1  1 


=■     to  find  X,  y,  and  2. 


8.  Given     -! 


Ans.  a;  =  12,    y  =  20,    2  =  30 

7a;  —  2;^  +  3m  =  17 
4y  —  20  +  t=  11 
5y  —  3a;  —  2ii  ■■=  8 
4y  —  3i«  +  2t  —  9 
3z  +  8m  =  33 
Ans.  a;  =  2,   y  =  4,   ^  =  3,    m  =  3,    t=-l. 


>     to    find    a;,    y,    z,    «, 
and  i. 


QUESTIONS. 


1.  What  fraction  is  that,  to  the  numerator  of  which,  if  1  be 
added,  its  value  will  be  one  third,  but  if  one  be  added  to  its  de 
nominator,  its  value   will  be  one  fourth. 


Let  the   fraction  be   represented  by    — . 

y 

a;  +  1         1  X  1 

Then,  by  the  question      =  -—    and    — -—  =  — -. 

y  3  y  +  1        4 

Whence  3a;  +  3  =  y,    and    4a;  =  y  +  1 . 

Therefore,  by  subtracting,      a?  —  3=1    or    a;=:4; 

and  3  X  4  +  3  =-  15  =  y. 

2.  A  market  woman  bought  a  certain  number  of  eggs  at  2  for 
a  penny,  and  as  many  more,  at  3  for  a  penny,  and  having  sold 
them  again  altogether,  at  the  rate  of  5  for  2d.,  found  that  she 
had  lost  4J. :  how  many  eggs  had  she  ? 


CHAP.  IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  87 

Let  2x  =     the  whole  number  of  eggs  ; 

then  X  =     the  number  of  eggs  of  each  sort ; 

and  -  X  :=     the  cost  of  the  first  sort ; 

2 

and  -  a;  =     the  cost  of  the  second  sort ; 

But  5  :  2a;  :  :  2  :  ^ ; 

5 

4x 
hence,     —     the  amount  for  which  the  eggs  were  sold. 
5 

Hence,  by  the  question, 

1  1  4x 

■*  —x-\ X =  4: 

2^3  5 

therefore  15a;  +  lOx  —  24x  =  120. 

Or,  X  =  120     the    number    of   eggs    of 

each  sort. 

3.  A  person  possessed  a  capital  of  30,000  dollars,  for  which  'la: 
drew  a  certain  interest  per  annum ;  but  he  owed  the  sum  o! 
20,000  dollars,  for  which  he  paid  a  certain  interest.  The  inter- 
est that  he  received  exceeded  that  which  he  paid  by  800  dollars. 
Another  person  possessed  $35,000,  for  which  he  received  interest 
at  the  second  of  the  above  rates  ;  but  he  owed  24,000  dollars, 
for  which  he  paid  interest  at  the  first  of  the  above  rates.  Tlu' 
interest  that  he  received  exceeded  that  which  he  paid  by  310 
dollars.     Required  the  two  rates  of  interest. 

Let  X  and  y  denote  the  two  rates  of  interest:    that  is,  the  in- 
terest of  SI 00  for  one  year. 

To  obtain  the  interest  of  $30,000  at  the  first  rate,  denoted  by  x, 

we  form  the  proportion 

30,000:r 
100  :   a;  :  :   30,000   :  :  — ——-    or    300a;. 

And  for  the  interest  $20,000,  the  rate  being  y, 

20,000y 
100  :  y  :  :  20,000  :  :   — ——^   or     200y. 
•^  100 

But  from  the  enunciation,  the  difference  between  these  two  in 

lerests  is  equal  to  800  dollars. 

We   have,  then,  for  the   first  equation  of  the   problem. 

300a:  —  200i/  =  800. 


SS  ELEMENTS    OF    ALGEBRA.  [CHAP.   IV. 

By  expressing  the  second  condition  of  the  problem  algebraically, 
we  obtain  the  other  equation, 

350y  —  240,x'  =  310. 

Both  members  of  the  first  e<iuation  being  divisible  by  100,  and 
iliose  of  the  second  by  10,  we  may  put  the  following,  in  plact; 
(if  them : 

3x  —  2y  =  8,        35y  —  24a:  -—  31 

To  eliminate  oc,  multiply  the  first  equation  by  8,  and  then  add 
i!   to  the  second ;    there  results 

19y  =  95,     whence    y  =  5. 

Substituting  for  y  its  value  in  the  first  equation,  this  equation 
becomes 

3a;  —  10  =  8,    whence    a;  =  6; 

Therefore,  the  first  rate  is  6  per  cent.,  and  the   second  5. 

Veri/icatM7i. 

$30,000,  placed  at  6  per  cent.,  gives      300  x  6  =   $1800. 
S20,000         do.         5  do.  200  X  5   =   $1000. 

And  we  have  1800  —  1000  =  800. 

The  second  condition  can  be  verified  in  the  same  manner. 
4.  There    are    three    ingots    formed   by    mixing    together    three 
metals  in  different  proportions. 

One  pound  of  the  first  contains  7  ounces  of  silver,  3  ounces  of 
copper,   and   6  ounces  of  pewter. 

One  pound  of  the  second  contains  12  ounces  of  silver,  3  ounces 
of  copper,  and   1    ounce  of  pewter. 

One  pound  of  the  third  contains  4  ounces  of  silver,  7  ounces 
of  copper,   and  5  ounces  of  pew"ter. 

It  is  required  to  form  from  these  three,  1  pound  of  a  fourth 
ingot  whicli  shall  contain  8  ounces  of  silver,  3|  ounces  of  copper, 
and  4^  ounces  of  pewter. 

Let      X  =     the  number  of  oinices  taken  from  the  first. 

y  =     the  number  of  ounces  taken  from  the  second. 
s  =     the  number  of  ounces  taken  from  the  third. 


CHAP.  IV.]  EQUATIONS    OF    THE    FIRST    DEGREK.  89 

Now,  since  1   pound  or   16  ounces  of  the  first  ingot  contains  7 

ounces    of  silver,   one    ounce  will   contain     —    of  7  ounces :    thai 

16 

is,     —    ounces  ;    and 
16 

7a; 
X   ounces  will  contain      -—     ounces  of  silver. 
Id 

12y 
y   ounces  will  contain     • — -    ounces  of  silver. 

^  16 

43 

z   ounces  will  contain     — -     ounces   of  silver. 
16 

But  since  1   pound  of  the  new  ingot  is  to  contain  8  ounces  of 

silver,  we  have 

Ix        12y       Az 

\ ^ =  8; 

16        16   ^  16 

or,  reducing  to  entire  terms, 

Ix  +  12y  +  40  =  128. 
For  the   copper,  3ar  +    3y  +  73^  =    60  ; 

and  for  the  pewter,     6a;  +       y  -f  5^  =     68. 

As  the  co-efficients  of  y  in  these  three  equations,   are  the  most 
simple,  we  will  eliminate  this  unknown   quantity  first. 

Multiplying  the   second  equation  by  4  and   subtracting  the  first, 
gives 

5a;  +  24^  =  112. 

Multiplying  the  third  equation  by  3  and  subtracting  the  second, 
gives 

15a;  +     8^  =  144. 

Multiplying  the  last  equation  by  3  and  subtracting  the  first,  gives 

40.r  =  320, 
whence  a;  =  8 

Substituting  this   value  of  x  in  the   equation 
5a;  +  24^=  112, 
it  becomes  40  +  24s  =  112,     whence    ^  =  3. 

Lastly,  the   two    values     a;  —  8    and    0=3,    being    substituK  d 
in  the  equation 

6x  +  y  +  50  =  68 

give  48  +  y  +  15  =  68,     whence     y  —  5 


90  ELEMENTS    OF    ALGEBRA  [CHAP.    IV. 

Therefore,  in  order  to  form  a  pound  of  the  fourth  ingot,  we 
must  take  8  ounces  of  the  first,  5  ounces  of  the  second,  and  3 
of  the  third. 

Vcnjication. 

If  there  be  7  ounces  of  silver  in  16  ounces  of  the  first  ingot, 
in  8  ounces   of  it,  there  should  be  a  number  of  ounces  of  silver 

expressed  by — . 

12  X  5               4x3 
In  like   manner,     and    —     wull    express    the    quan- 
tity of  silver   contained   in  5  ounces  of  the   second   ingot,  and   3 
ounces  of  the  third.     Now,  we  have 

7X8        12  X  5       4x3         128 


16  16  16  16 

therefore,  a  pound  of  the  fourth  ingot  contains  8  ounces  of  silver, 
as  required  by  the  enunciation.  The  same  conditions  may  be  veri- 
fied relative  to  the  copper  and  pewter. 

5.  What  two  numbers  are  those,  whose  difference  is  7,  and 
sum  33  ?  Ans.  13   and  20. 

6.  To  divide  the  number  75  into  two  such  parts,  that  three 
limes  the   greater  may  exceed  seven  times  the  less   by  15. 

Ans.  54   and  2]. 

7.  In  a  mixture  of  wine  and  cider,  %  of  the  whole  plus  25  gal- 
lons was  wine,  and  ^  part  minus  5  gallons  was  cider  ;  how  many 
gallons  were   there  of  each  ? 

Ans.  85  of  wine,  and  35  of  cider. 

8.  A  bill  of  £120  was  paid  in  guineas  and  moidores,  and  the 
number  of  pieces  of  both  sorts  that  were  used  was  just  100;  if 
the  giunca  were  estimated  at  21.?.,  and  the  moidore  at  27^.,  how 
many  were   there   of  each  1  Ans.  50  of  each. 

9.  Two  travellers  set  out  at  the  same  time  from  London  and 
York,  whose  distance  apart  is  150  miles  ;  one  of  them  goes  8 
miles  a  day,  and  the  other  7 ;   in  what  time  will  they  meet  ^ 

Ans.  In  10  days, 

10.  At  a  certain  election,  375  persons  voted  for  two  candi 
dates,  and  the  candidate  chosen  had  a  majority  of  91  ;  how  inuiiy 
voted   for  each?  Ans.  233  for  one,  and  142  for  the  other 


CHAP.  IV.]  Ey.UATIONS    OF    THE    FIRST    DEGREE.  91 

11.  A's  age  is  double  of  B's,  and  B's  is  triple  of  C's,  and  the 
sum  of  all  their  ages  is   140;    what  is  the  age  of  each? 

Ans.  A's  =  84,    B's  =  42,    and    C's  =  14. 
12    A  person   bought   a   chaise,  horse,  and  harness,    for  jG60  ; 
the  horse  came  to  twice  the  price  of  the  harness,  and  the  chaise 
to  twice   the  price  of  the  horse  and  harness  ;    what  did  he  give 
for  each?  ^  jC13     6s.  8d.     for  the  horse. 

Ans.    <  jG  6   13.y.  4:d.     for  the  harness. 
C  ;£40  for  the  chaise. 

13.  Two  persons,  A  and  B,  have  both  the  same  income.  A 
saves  1  of  his  yearly ;  but  B,  by  spending  £50  per  annum  more 
than  A,  at  the  end  of  4  years  finds  himself  £100  in  debt;  what 
is   the  income  of  each  ?  Ans.  £125. 

14.  A  person  has  two  horses,  and  a  saddle  worth  £50;  now, 
if  the  saddle  be  put  on  the  back  of  the  first  horse,  it  w^ill  make 
his  value  double  that  of  the  second  ;  but  if  it  be  put  on  the  back 
of  the  second,  it  will  make  his  value  triple  that  of  the  first ; 
what  is  the  value  of  each  horse  ? 

Ans.  One  £30,  and  the  other  £40. 

15.  To  divide  the  number  36  into  three  such  parts,  that  ^  of 
the  first,  J  of  the  second,  and  ^  of  the  third,  may  be  all  equal  to 
each  other.  Ans.  8,   12,  and  16. 

16.  A  footman  agreed  to  serve  his  master  for  £8  a  year  and 
a  livery,  but  \vas  turned  away  at  the  end  of  7  months,  and  re- 
ceived only  £2   13^.  4  J.  and  his   livery  ;   what  was  its  value  ? 

}  Ans.  £4  16s. 

17.  To  divdde  the  number  90  into  four  such  parts,  that  if  the 
first  be  increased  by  2,  the  second  diminished  by  2,  the  third 
multiplied  by  2,  and  the  fourth  divided  by  2,  the  sum,  difference, 
product,  and  quotient  so  obtained,  will  be  all  equal  to  each  other. 

Ans.  The  parts  are   18,  22,  10,  and  40. 

18.  The  hour  and  minute  hands  of  a  clock  are  exactly  together 
at  12  o'clock ;    when  are  they  next  together  ? 

Ans.  1  h.  5^Y  i^i'^- 

19.  A  man  and  his  wife  usually  drank  out  a  cask  of  beer  in 
12  days;  but  when  the  man  was  from  home,  it  lasted  the  woman 
30  days ;  how  many  days  would  the  man  be  in  drinking  it  alone  ? 

Ans.  20  days. 


92  ELEMENTS    OF    ALGEBRA.  [CliAP.    'V. 

20.  If  A  and  B  together  can  perform  a  piece  of  v/ork  in  8 
days,  A  and  C  together  in  9  days,  and  B  and  C  in  10  days  ; 
how  many  days  would  it  take  each  person  to  perform  the  same 
work  alone?  Ans.  A   14||  days,    B   17  |f,  and  C  23  j^j. 

21.  A  laborer  can  do  a  certain  work  expressed  by  a,  in  a  time 
expressed  by  6 ;  a  second  laborer,  the  work  c  in  a  time  d ;  a 
third,  the  work  e  in  a  time  /.  Required  the  time  it  would  take 
the   three    laborers,  working  together,   to    perform  the  work  g. 

Ans.  X  =  —- ~ —- . 

adj  +  ocf  -j-  ode 

Application. 

a  =  27 ;    J  =  4  |  c  =  35  ;    (^  =  6  ]  e  :=  40  ;  /=  12  |  ^  =  191  ; 

X  will  be  found  equal  to   12. 

22.  If  32  pounds  of  sea  water  contain  1  pound  of  salt,  how 
much  fresh  water  must  be  added  to  these  32  pounds,  in  order 
that  the  quantity  of  salt  contained  in  32  pounds  of  the  new  mix 
ture   shall  be  reduced   to   2   ounces,   or  -1   of  a  pound  ? 

Ans.  224  lbs. 

23.  A  number  is  expressed  by  three  figures  ;  the  sum  of  these 
figures   is    11  ;   the   figure  in  the  place   of  units  is  double  that  in 
the  place  of  hundreds  ;    and  when  297   is   added  to  this  number, 
the  sum  obtained  is   expressed  by  the   figures   of  this  number  re 
versed.     What  is  the  number  ?  Ans.  326. 

24.  A  person  who  possessed  100,000  dollars,  placed  the  greater 
part  of  it  out  at  5  per  cent,  interest,  and  the  other  part  at  4  per 
cent.  The  interest  which  he  received  for  the  whole  amounted 
to  4640   dollars.     Required  the   two   parts. 

Ans.  $64,000    and    $36,000. 

25.  A  person  possessed  a  certain  capital,  which  he  placed  out 
at  a  certain  interest.  Another  person  possessed  10,000  dollars 
more  than  the  first,  and  putting  out  his  capital  1  per  cent,  more 
advantageously,  had  an  income  greater  by  800  dollars.  A  third, 
possessed  15,000  dollars  more  than  the  first,  and  putting  out  his 
capital  2  per  cent,  more  advantageously,  had  an  income  greater 
by  1500  dollars.  Required  the  capitals,  and  the  three  rates  of 
interest. 

Sums  at  interest,       $30,000,    $40,000,     $45,000. 

Rates  of  interest,  4  5  6        per  ceni 


CHAP.  IV.]  EQUATION'S    OF    THE    FIRST    DEGREF,.  93 

26.  A  banker  has  two  kinds  of  money  ;  it  takes  a  pieces  of 
the  first  to  make  a  crown,  and  h  of  the  second  to  make  the 
same  sum.  Some  one  offers  him  a  crown  for  c  pieces.  How 
many  of  each  kind  must  the  banker  give  him  ? 

A         ■,       11      a(c  —  b)       ^  ,  ,  .    ,      b  (a  —  c) 

Afis.   1st  kind,     — ^ -^  ;     2d  kind,    — ^ ~. 

a  —  o  a  —  0 

27.  Find  what  each  of  three  persons,  A,  B,  C,  is  worth,  know 
ing,   1st,  that  what   A  is    worth   added   to   I  times    what   B    and    C 
are   worth,  is   equal  to  p ;    2d,   that  what  B  is   worth  added  to  m 
times  what  A  and  C  are  worth,  is  equal  to  q  ;  3d,  that  what  C  is 
worth  added  to  n  times   what  A   and  B  are   worth,  is  etjual  to  r. 

If  we  denote  by  s  what  A,  B,  and  C,  are  worth,  we  introduce 
into  the  calculus  an  auxiliary  unknown  quantity,  and  resolve  the 
question  in  a  very  simple  manner.  The  term  calculus,  in  its  gen- 
eral sense,  denotes  any  operation  performed  on  algebraic  quantities. 

28.  Find  the  values  of  the  estates  of  six  persons.  A,  B,  C,  D, 
E,  F,  from  the  following  conditions :  1st.  The  sum  of  the  estates 
of  A  and  B  is  equal  to  a ;  that  of  C  and  D  is  equal  to  b  ;  aud 
that  of  E  and  F  is  equal  to  c.  2d.  The  estate  of  A  is  worth  m 
times  that  of  C  ;  the  estate  of  D  is  worth  n  times  that  of  E,  and 
the  estate  of  F  is  worth  p  times  that  of  B. 

This  problem  may  be  resolved  by  means  of  a  single  equation, 
involving  but  one  unknown  quantity. 

Explmmtion  of  Negative  Results. 

104.  The  algebraic  signs  are  an  abbreviated  language.  They 
indicate  certain  operations  which  are  to  be  performed  on  the  quan- 
tities before  which  they  are   placed. 

The  operation  indicated  by  a  particular  sign,  must  be  per- 
formed on  CA'ery  quantity  before  which  the  sign  is  placed.  In- 
deed, the  principles  of  Algebra  are  all  established  upon  the 
supposition,  that  each  particular  sign  which  is  employed  means 
always  the  same  thing  ;  and  that  whatever  it  requires  is  strictly 
performed.  Thus,  if  the  sign  of  a  quantity  is  +»  "^ve  understand 
that  the  quantity  is  to  be  added ;  if  the  sign  is  — ,  we  under- 
stand that  it  is  to  be  subtracted. 

For  example,  if  we  have  —  4,  it  indicates  that  this  4  is  to 
be  subtracted  from  some  other  number,  or  that  it  is  the  result  of 
a   subtraction  but  partially  made. 


94  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV 

If  it  were  required  to  subtract  20  from  16,  the  subtraction  could 
not  be  made  by  the  rules  of  arithmetic,  since  20  is  greater  than  16. 
By  observing  that 

20  ::=  16+  4, 

we   may  express   the  subtraction  thus, 

16  —  20  =1  16  —  16  —  4  =:  —  4. 

We  thus  make  the  subtraction  of  20  from  16  as  far  as  it  is 
possible,  and  obtain  a  remainder  4  with  a  minus  sign,  which  in- 
dicates that  4   is  still  to  be  treated  as   a  subtractive  quantity. 

To  show  the  necessity  of  giving  to  this  remainder  its  proper 
sign,  let  us  suppose  that  10  is  to  be  added  to  the  difference  of 
1 6  —  20  ;  or  what  is  the  same  thing,  that  20  is  to  be  subtracted 
from  26. 

The  numbers  would  then  be  written 
16  —  20=  —    4 
+  10  =  +  10 


26  —  20  =  +     6; 

and  had  the  —  sign  not  been  preserved  in  the  first  subtraction, 
the  second  result  would  have   been   +14  instead  of  +  6. 

105.  If  the  sum  of  the  negative  quantities  in  the  first  member 
of  the  equation,  exceeds  the  sum  of  the  positive  quantities,  the 
second  member  of  the  equation  will  be  negative,  and  the  verifica- 
tion of  the   equation   will   show  it  to  be   so. 

For  example,  if  a  —  b  =  c, 

and  we  make       a  =  15    and    b  =  18,    c  will  be    =  —  3. 

Now,  the  essential  sign  of  c  is  different  from  its  algebraic  sign 
.n  the  equation.  This  arises  from  the  circumstance,  that  the 
equation 

a  —  b  =  c 

expresses  generally,  the  difference  between  a  and  b,  without  in- 
dicating which  of  them  is  the  greater.  When,  therefore,  we  at- 
tribute particular  values  to  a  and  5,  the  sign  of  c,  as  well  as  its 
vahin,  becomes   known. 

We   will  illustrate  these   remarks  bj'-  a   few   examples. 


CHAP.  IV.]  EQUATIONS    OF    TUV.    KIRST    DEGREE.  95 

1.  To  find  a  number,  whiclx  added  to  the  number  b,  will  give 
a  sum  equal  to  the  number  a. 

Let  X  =   the  required  number. 
Then,  by  the   conditions 

X  -{-  b  ^  a,    whence    x  =  a  —  b. 

This  expression,  or  formula,  will  give  the  algebraic  value  of 
r  iu  all  the  particular  cases  of  this  problem. 

For  example,  let        a  =  47    and    6  =  29  ; 

then,  a;  =  47  — 29  =  18. 

Again,  let  c  =  24    and    b  —  3}  ; 

then,  a;  =  24  —  31  =  —  7. 

This  last  value  of  a;,  is  called  a  negative  solution.  How  is  it 
to  be  interpreted? 

If  we  consider  it  as  a  purely  arithmetical  result,  that  is,  as 
arising  from  a  series  of  operations  in  which  all  the  quantities  are 
regarded  as  positive,  and  in  which  the  terms  add  and  subtract 
imply,  respectively,  augmentation  and  diminution,  the  problem  will 
obviously  be  impossible  for  the  last  values  attributed  to  a  and  b ; 
for,  the  number  b  is   already  greater  than   24. 

Considered,  however,  algebraically,  it  is  not  so ;  for  we  have 
found  the  value  of  a;  to  be  —  7,  and  this  number  added,  in  the 
algebraic  sense,  to  31,  gives  24  for  the  algebraic  sum,  and  there- 
fore  satisfies  both  the   equation  and  enunciation. 

2.  A  father  has  lived  a  number  a  of  years,  his  son  a  number 
of  years  expressed  by  b.  Find  in  how  many  years  the  age  of 
the  son   will  be   one  fourth   the  age  of  the   father. 

Let  X  =    the  required  number  of  years. 

Then  a  -\-  x  =    the  age  of  the  father  >   at  the  end  of  the  re- 

and  b  -\-  X  =    the  age  of  the  son      )       quired  time. 

Hence,  by  the  question 

a  -{-  X  a  —  4h 

— : —  =  b  -\-  x:     whence,     x  = . 

4  '  3 

o  =.^  1    I       n     .1,  54  -  36       18 

buppose     a  —  54,    and    6  =  9;   then  x   = =  —  =  (J. 

"  3  3 


06  ELEMENTS    OF    ALGEBRA.  [CHAP.   IV. 

The  father  being  54  years  old,  and  the  son  9,  in  6  years  the 
father  will  be  60  years  old,  and  his  son  15;  now  15  is  the 
tDurili  of  60  ;    hence,   a;  =  6    satisfies  the  enunciation. 

Let  us  now  suppose     a  =  45,    and    i  =:  15  ; 

45  —  60 
then,  X  = =  —  5. 

If  we  substitute  this  value  of  x  in  the  equation  of  condition, 

b  -\-  X, 


4       -'^    ' 

we  obtain, 
and 

'^^  -  '  -  15 

4        -^^ 
10  =  10, 

Hence,  —  5  substituted  for  x,  verifies  the  equation,  and  therefore 
js  a  true  answer. 

Now,  the  positive  result  which  was  obtained,  shows  that  the 
age  of  the  father  will  be  four  times  that  of  the  son  at  the  ex- 
pirn,tioii  of  6  years  from  the  time  when  their  ages  were  con- 
sidered ;  while  the  negative  result,  indicates  that  the  age  of  the 
father  was  four  times  that  of  his  son,  5  years  previous  to  the 
time  when  their  ages  were  compared. 

The  question,  taken  in  its  general,  or  algebraic  sense,  demands 
the  time,  at  which  the  age  of  the  father  was  four  times  that  of 
the  son.  In  stating  it,  we  supposed  that  the  age  of  the  father 
was  to  be  augmented ;  and  so  it  was,  by  the  first  supposition.  But 
the  conditions  imposed  by  the  second  supposition,  required  the 
age  of  the  father  to  be  diminished,  and  the  algebraic  result  con- 
formed to  this  condition,  by  appearing  with  a  negative  sign.  If 
we  wished  the  result,  under  the  second  supposition,  to  have  a 
positive  sign,  we  might  alter  the  enunciation  by  demanding,  how 
many  years  since  the  age  of  the  father  was  four  times  that  of  the 
son. 

U  X  =  the  number  of  years,  we  shall  have 

a  —  X       ,  .  4b  —  a 

=:  6  —  X  :     hence,    ,t  =  — . 

4  3 

If  a  =  45    and    h  —  15,    x  will  be   equal   to   5. 

Reasoning    from    analogy,    we    establish    the    following    general 

principles, 


CHAP.  IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  97 

Ist.  Every  negative  value  found  for  the  unknown  qua7itity  in  a 
problem  of  the  first  degree,  will,  when  taken  with  its  proper  sign, 
verify  the   equation  from  which  it  was  derived. 

2d.  That  this  negative  value,  taken  with  its  proper  sig7i,  will  also 
satisfy  the  enunciation  of  the  problem,  understood  in  its  algebraic 
sense. 

3(1.  The  negative  result  shows  that  the  enunciation  is  impossible 
legarded  in  its  arithmetical  sense.  The  language  of  Algebra  de- 
tects the  error  of  the  arithmetical  enunciation,  and  hidicates  the  gen- 
eral relation  of  the  quantities. 

4th.  The  negative  result,  considered  without  reference  to  its  sign, 
may  be  regarded  as  the  answer  to  a  problem  of  which  the  enuncia- 
tion only  differs  from  the  one  proposed  in  this :  that  certain  quan- 
tities which  were  additive  have  become   subtractive,  and  reciprocally 

106.  As  a  further  ilhistration  of  the  change  which  an  alge- 
braic sign  may  produce  in  the  enunciation  of  a  problem,  let  us 
resume  that  of  the  laborer  (page  76). 

Under  the  supposition  that  the  laborer  receives  a  sum  c,  we 
have  the  equations 

X  -\-    y  :=z  n^  ,  hn  -{-  c  an  —  c 

^  \      whence,      x= — -,       y~ -— . 

ax  —  by  ^  c  )  a  -\-  b  a  -\-  b 

If  at  the  end  of  the  time,  the  laborer,  instead  of  receiving  a 
sum  c,  owed  for  his  board  a  sum  equal  to  c,  then,  by  would  be 
greater  than  ax,  and  under  this  supposition,  we  should  have  the 
equations 

a?  +  y  ==  ra    and    ax  —  by  =  —  c. 

Now,  it  is  plain  that  we  can  obtain  immediately  the  values  of 
X  and  y,  in  the  last  equations,  by  merely  changing  the  sign  of 
c    in    each  of  the    values    found    from   the    equations   above ;    this 

gives 

In  —  c  an  -\-  e 

The  results  for  both  enunciations,  may  be  compreiiended  in  the 
same  formulas,  by  writing 

bn  ±  c  an  ^  c 

X  =  r  ;  y  =:   — —r- 

a  +  b  -^  a  +  b 

The  double  sign  ±,  is  read  plus  or  minus,  and  q:,  is  read,  rni 
nus  or  plus.     The  upper    signs    correspond  to   the   case  in   wliich 


98  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV 

the  laboier  received,   and  the   lower  signs,  to  the  case  in  which 
he  owed  a  sum   c.     These  formulas  also  comprehend  the  case  in 
which,   in   a   settlement   between   the    laborer    and   his   employer, 
their  accounts  balance.     This  supposes   c  =  0,    which  gives 
bn  an 

^  =  7+T'      ^  =  7+1' 

Discussion  of  Problems.     Explmiat'ion  of  the  terms 
Nothing  a7id  lujinity. 

107.  When  a  problem  has  been  resolved  generally,  that  is,  by 
means  of  letters  and  signs,  it  is  often  required  to  determine  what 
the  values  of  the  unknown  quantities  become,  when  particular  sup- 
positions are  made  upon  the  quantities  which  are  given.  The 
determination  of  these  values,  and  the  interpretation  of  the  pe- 
culiar results  obtained,  form  what  is  called  the  discussion  of  the 
problem. 

The  discussion  of  the  following  question  presents  nearly  all 
the  circumstances  which  are  met  with  in  problems  of  the  first 
degree. 

108.  Two  couriers  are  travelling  along  the  same  right  line  and 
in  the  same  direction  from  R^  toward  R.  The  number  of  miles 
travelled  by  one  of  them  per  hour  is  expressed  by  m,  and  the 
number  of  miles  travelled  by  the  other  per  hour,  is  expressed 
by  n.  Now,  at  a  given  time,  say  12  o'clock,  the  distance  be- 
tween them  is  equal  to  a  number  of  miles  expressed  by  a :  re- 
quired the  time  when  they  will  be  together. 

R^ A B R. 

At  12  o'clock  suppose  the  forward  courier  to  be  at  B,  the  other 
at  A,  and  R  to  be  the  point  at  which  they  will  be  together. 
Then,     AB  =  a,  their  distance   apart  at  12  o'clock. 
Let  t  =        the  number  of  hours  which  must  elapse,  be- 

fore they  come  together  ; 
and  X  =        the  distance  BR,  which  is  to  be  passed  over 

by  the   forward  courier. 
Then,   since   the   rate   per   hour,  multiplied  by  the   number  of 
hours    will  give   the   distance   passed  over  by  each,   we   have, 
t  X  m  =z  a  -\-  X  ~  AR 
t  X    n  zz:  ic  —  BR 


■JHAP.  IV.]  EQUATIONS    OF    THE    FIRST    DEGREE.  99 

Hence  by  subtracting, 

t  (m  —  n)  =z  a, 

and  hence,  t  = . 

m  —  n 

Now,  so  long  as  m  '^  n,  t  will  be  positive,  and  the  problem 
will  be  solved  in  the  arithmetical  sense  of  the  enunciation.  For 
if  771  >  71,  the  courier  from  A  will  travel  faster  than  the  courier 
from  B,  and  will  therefore  be  continually  gaining  on  him :  the 
interval  which  separates  them  will  diminish  more  and  more,  un- 
til it  becomes  0,  and  then  the  couriers  will  be  found  upon  the 
same  point  of  the  line. 

In  this  case,  the  time  t,  which  elapses,  must  be  added  to  12 
o'clock,  to  obtain  the  time  when  they  are  together. 

But,  if  we  suppose  m  <i  n,  then,  m  —  n  will  be  negative,  and 
the  value  of  t  will  be  negative.  How  is  this  result  to  be  inter- 
preted 1 

It  is  easily  explained  from  the  nature  of  the  question,  which 
considered  in  its  most  general  sense,  demands  the  time  when  the 
couriers  are  together. 

Now,  under  the  second  supposition,  the  courier  which  is  in  ad- 
vance, travels  the  fastest,  and  therefore  will  continue  to  separate 
himself  from  the  other  courier.  At  12  o'clock  the  distance  be- 
tween them  was  equal  to  a:  after  12  o'clock  it  is  greater  than  a; 
and  as  the  rate  of  travel  has  not  been  changed,  it  follows  that 
previous  to  12  o'clock  the  distance  must  have  been  less  than  a. 
At  a  certain  hour,  therefore,  before  12,  the  distance  between  them 
must  have  been  equal  to  nothing,  or  the  couriers  were  together  at 
some  point  R^.  The  precise  hour  is  found  by  subtracting  the 
value  of  t  from    12   o'clock. 

This  example,  therefore,  conforms  to  the  general  principle,  that, 
if  the  conditions  of  a  problem  are  such  as  to  render  the  vnknovm 
quantity  essentially  negative,  it  will  appear  in  the  result  loith  the 
minus  sign,  whenever  it  has  been  regarded  as  positive  in  the  enun- 
ciation. 

If  we  wish  to  find  the  distances  AR  and  BR,  passed  over  by 
the  two  couriers  before  coming  together,  we  may  take  the  equation 

a 
t=  


m  — 

n 

na 

m  — 

n 

ma 

m  — 

n 

na 

100  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV 

and  multiply  both  members  by  the   rates  of  travel  respectively : 
this  will  give 

AR  z=mt  = and 


BR  =  nt  = 

Also,  AR'  =  —  mt  — 

and  BR'  =  —  n?  = 

m  —  n 

from  which  we  see,  that  the  two  distances  AR  and  BR,  will 
both  be  positive  when  estimated  toward  the  right,  and  that  AR' 
and  BR'  will  both  be  negative  when  estimated  in  the  contrary 
direction. 

109.  To  explain  the  terms  nothing  and  infinity,  let  us  considei 

the   equation 

a 

t  —  . 

m  —  n 

If  the  couriers  travel  at  different  rates,  m  —  n  will  be  a  finite 
quantity,  and  its  sign  will  depend  on  the  relative  values  of  m 
and  n.     Designate  this  quantity  by  A. 

Now,  if  we  suppose    a  =  0,   we  shall  have 

«  =  — -  ;    or    <  X  A  =:  0  ; 
A 

an  equation  which  can  only  be   satisfied  by  making   t  =.0. 

To   interpret  this  result,  let  us   go  back  to  the   enunciation  of 

the   problem.     If  a  =  0,    the  couriers   are  together  at  12  o'clock ; 

and  since   they  travel  at  different  rates,  they  can  never  be  again 

together :    hence,   t  can  have  no   other  value  than  0.     Therefore, 

we  conclude  that,  the  quotient  of  0  divided  by  a  finite  quantity,  is  0. 

110.  Let  us  resume  the  equation 

a 

t  = . 

m  —  n 

If  in  this  equation  we  make  m  =:  n,  then  m  —  n  :=  0,  and 
the  value   of  t  will  reduce   to 

i  =  -TT    or    ^  X  0  =  a ; 
an   equation  which  cannot  be   satisfied  for  any  finite   value   of  t. 


CHAP.  IV.]  NOTHING    AND    INFINITY.  101 

In  order  to  interpret  this  new  result,  let  us  go  back  to  the 
enunciation  of  the  question.  We  see  at  once,  that  it  is  abso- 
lutely impossible  to  satisfy  the  enunciation  for  any  finite  value 
for  t ;  for,  whatever  time  we  allow  to  the  two  couriers,  they  can 
never  come  together,  since  being  once  separated  by  an  inter- 
val a,  and  travelling  equally  fast,  this  interval  will  always  be  pre- 
served. 

a 

Hence,  the  result,  -—  may  be  regarded  as  a  sign  of  impossi- 
bility for  any  finite  value  of  t. 

Nevertheless,  algebraists  consider  the  result, 

a 

'  =  ¥• 

as    forming    a   species    of  value,    to    which   they  have    given    the 

name  of  infinite  value,   for  this   reason : 

When  the   difference  m  —  n,  without  being  absolutely  nothing, 

is  supposed  to  be  very  small,  the  result 

a 

t  = . 

m  —  n 

is  very  great. 

Take,  for  example,  m  —  n  =  0,01. 

Then  t  =  — - —  =  — ^  =  100a. 

m  —  n        0.01 

Again,  take        ^  m  —  n  =  0.001,    and  we  have 

a  a 


-  1000a. 


m  —  n        0.001 

In  short,  if  the  difference  between  the  rates  is  not  zero,  the 
couriers  will  come  together  at  some  point  of  the  line,  and  the 
time  will  become  greater  and  greater,  as  this  difference  is  dimin- 
ished. 

Hence,  from  analogy,  if  the  difference  between  the  rates  is  less 
than  any  assignable  number,  the  time  expressed  by 

a  a 

m  —  n         0  ' 
vnll   be  greater  than   any.  assignable  or  fimite   number.     Therefore 
for  brevity,  we   say,  when    m  —  n  =  0,    the  result, 


102  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV. 

becomes    equal   to    infinity,   wliich    we    designate   by   the    charac- 
ter   00. 

Hence  we  conclude,  that  a  finite  quantity  divided  by  0,  gives 
a    quotient    greater    than    any    assignable    quantity,    which    we    call, 

IM'INITV. 

111.  Again,  let  A  represent  any  finite  number :  then,  since  the 
vahie  of  a  fraction  increases  as  its  numerator  becomes  greater 
with  reference  to  its  denominator,   the   expression 

A 

O"' 
is   a  proper  symbol  to  represent    an    infinite  quantity ;    that   is,  a 
quantity  greater  than  any  assignable  quantity. 

Since  the  value  of  a  fraction  diminishes  as  its  denominator  be 
comes  greater  with  reference  to  its  numerator,  the  expression 

A_ 

00 

is  a  proper  symbol  for  a  quantity  less   than  any  assignable   quan 
tity.     Hence, 

—  and    00 
0 

are  synonymous  symbols  ;  and  so  likewise,  are 

—  and    0. 

00 

We  have  been  thus  particular  in  explaining  these  ideas  of  in- 
finity, because  there  are  some  questions  of  such  a  nature,  that 
infinity  may  be  considered  as  the  true  answer  to  the  enunciation. 

In  the  case  just  considered,  where  m  —  n,  it  will  be  perceived 
that  there  is  not,  properly  speaking,  any  solution  in  finite  and  dc' 
terminate  numbers  ;  but  the  value  of  the  unknown  quantity  is  found 
to  be  infinite. 

112.  If,  in  addition  to  the  hypothesis  m  =  n,  we  also  suppose 

that  a  =  0,  we  have  t  =  — ,  or  i  X  0  =  0 ;    a  result    which    will 

be  satisfied  by  any  value  of  t. 

To  interpret  this  result,  let  us  consider  again  the  enunciation, 
from  which  it  is  perceived,  that  if  the  two  couriers  travel  equally 
fast,  and  are  once  at  the  same  point,  they  ought,  ever  after,  to 
be   together,  and   consequently  the  required  time  is  entirely  undo- 


OHAP.  IV.]  NOTHING    AND    INFINITY  103 

tenninecl.      Therefore,   the    expression    — -     is,  in   this    case,   the 

symbol  of  an  indeterminate  quantity. 

The  preceding  suppositions  are  the  only  ones  that  lead  to  re- 
markable results  ;  and  they  are  sufficient  to  show  to  beginners 
the  manner  in  which  the  results  of  Algebra  answer  to  all  the  cir- 
cumstances  of  the   enunciation  of  a  problem. 

113.  It  should  be   observed,  that   the   expression    —    is   not   a 

certain   symbol   of  indetermination,  but  frequently  arises   from    the 

existence  of  a  common  factor  in   each  term  of  the  fraction,  which 

factor  becomes  nothing,  in  consequence  of  a  particular  hypothesis. 

For  example,  suppose  the  value  of  the  unknown  quantity  to  be 

_  a3  -  b'^ 

If,  in  this   formula,   a  is  made   equal  to   h,  there   residts 

0 
0 
But  observe  (Art.  48),  that 

a^  -h^  z^{a~  b)  (a2  +  afi  -f  ^2) 
and  a^  -b"^  =  {a  —  h)  {a  +  b), 

hence,  we  have  ' 

^{^a-h)  (a2  +  oJ  4-  h-^) 
*"  ~         {a  ~h){a  +  b)       ' 
Now,  if  we  suppress  the  common  factor   a  —  b,   and  then  sup- 
pose   a  =  b,    we  shall  have 

_   a2  -f  ff^  +  b'^  _   3a2  ^  3a 
a  -\-  b  2a         2 

Let  us  suppose,  that  in  another  example,  we  have 
,  a2  _  52 


X  =z 


{a  -  6)2 

If  we  suppose   a  z:^  b,   we  have 

0 


0 


If,  liowever,  we   suppress   the   factor  common  to   the  numerator 
and  denominator,  in  the  value  of  x,  we  have, 

_{a  +  b)  {a  —  b)  _  a  -\-  b  _2b  _ 
~  (a  —  b)[a  —  b)  ~  a  —  b  ~  "o"  ~  °°* 


104  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV 

Therefore  we  conclude,  that  before  pronouncing  upon  the  true 
vahie  of  the   fraction, 

"o" 

it  is   necessary  to  ascertain  whether  the  two  terms  do  not  contain 
a  common  factor.     If  they  do  not,  we  conclude  that  the  fraction  is 
really  indeterminate.     If  they  do  contain  one,  suppress  it,  and  then 
make  tlie  particular  hypothesis  ;    this  will  give  the  true  value  of 
the   fraction,  which  will  assume  one   of  the  three   forms 
A^      A        0       ' 
'B'    "o"'    ¥' 
that  is,   it  will   be   determinate,  infinite,  or  indeterminate. 

This  observation  is  very  useful  in  the- discussion  of  problems. 

Of  IncqualUics. 

114.  In  the  discussion  of  problems,  Ave  have  often  occasion  to 
suppose  quantities  unequal,  and  to  perform  transformations  upon 
them,  analogous  to  those  executed  upon  equalities.  We  some- 
times do  this,  to  establish  the  necessary  relations  between  the 
given  quantities,  in  order  that  the  problem  may  be  susceptible  of 
a  direct,  or  at  least,  of  a  real  solution.  We  often  do  it,  to  fix  the 
limits  between  which  the  particular  values  of  certain  given  quan- 
tities must  be  found,  in  order  that  the  enunciation  may  fulfil  a 
particular  condition.  Now,  although  the  principles  established  for 
equations  are,  in  general,  applicable  to  inequalities,  there  are  nev- 
ertheless some  exceptions,  of  which  it  is  necessary  to  speak,  in 
order  to  put  the  beginner  upon  his  guard  against  some  errors  that 
he  might  commit,  in  making  use  of  the  sign  of  inequality.  These 
exceptions  arise  from  the  introduction  of  negative  expressions  into 
the  calculus,  as  quantities. 

In  order  to  be  clearly  understood,  we  will  give  examples  of 
the  different  transformations  to  which  inequalities  may  be  sub- 
jected, taking  care  to  point  out  the  exceptions  to  which  these 
transformations  are  liable. 

115.  Two  inequalities  are  said  to  subsist  in  the  same  sense, 
when  the  greater  quantity  stands  at  the  left  in  both,  or  at  the 
right  in  both  ;  and  in  a  contrary  sense,  when  the  greater  quan- 
litv  stands  at  the  right  in   one,   and   at  the  left  in  the   other. 


CHAP.   IV. J  OF    IXliQUALlTIES.  105 

Thus,       25>20     and     18  >  10,    or     6<8    and    7  <  9, 

are   inequalities   which  subsist   in    the    same   sense ;    and   the    in- 
equalities 

15  >  13    and     12  <  14, 

subsist  in  a  contrary  sense. 

\.  If  we  add  the  same  quantity  to  both  members  of  un  inequality, 
or  sxibtract  the  same  quantity  from  both  members,  the  resulting  in- 
equality will  subsist  in  the  same  sense. 

Thus,  take    8  >  6  ;    by  adding  5,  we  still  have 
8  +  5>6+5; 
and  subtracting  5,  we  have 

8  —  5  >  6  —  5. 

When  the  two  members  of  an  inequality  are  both  negative,  that 
one  is  the  least,  algebraically  considered,  which  contains  the  great- 
est number  of  units.  Thus,  —  25  <  —  20  ;  and  if  30  be  added 
to  both  members,  we  have  5  <|  10.  This  must  be  understood 
entirely  in  an  algebraic  sense,  and  arises  from  the  convention  be- 
fore established,  to  consider  all  quantities  preceded  by  the  minus 
sign,   as  subtractive. 

The  principle  first  enunciated,  serves  to  transpose  certain  terms 
from  one  member  of  the  inequality  to  the  other.  Take,  for  ex- 
ample, the   inequality 

a2  +  Z,3  >  3b^^  —  2a2 ; 
there  will  result,  by  transposing, 

a3  +  2a2  >  3/;2  _  i2^    or    3a2  >  2J2. 

2.  Jf  two  inequalities  subsist  in  the  same  sense,  and  we  add  them 
member  to  member,  the  resulting  inequality  will  also  subsist  in  the 
same  sense. 

Thus,  add  o  !>  ^,    c  >  J,    e  y>  f:     and 

there  results  a-\-c-{-e^b-{-d-\-f. 

But  this  is  not  always  the  case,  when  we  subtract,  member  from 
member,   two  inequalities   established  in   the   same  sense. 

Let  there  be  the  two  inequalities    4  <  7    and   2  <  3,  we  have 
4  —  2    or    2  <  7  —  3    or    4. 


106  ELEMENTS    OF    ALGEBRA.  [CHAP.  IV. 

But  if  we  have  the  inequalities  9  <  10  and  6  <  8,  by  sub- 
tracting we  have 

9  —  6   or    3  >  10  —  8   or   2. 

We  should  then  avoid  this  transformation  as  much  as  possible, 
or  if  we  employ  it,  determine  in  which  sense  the  resulting  in- 
equality exists. 

3.  If  the  two  members  of  an  inequality  he  multiplied  hy  a  positive 
number,   the  resulting  inequality  will  exist  in  the  same   sense. 

Thus,  a  <  ^,    will   give    3a  <  3J  ; 

and,  —  a  <  —  i,     —  3a  <  —  35. 

This  principle   serves   to  make   the   denominators   disappear, 

„  ,       .  ,.        a"^  —  h"^       c^  —  d^  T    ,  ,  , 

From   the  mequality —  >  - — ,    we   deduce,   by   mul- 

2a  6a 

tiplying  by  6a^, 

3a  (a2  -  i2)  >  2d  (c^  -  J2), 

and  the   same   principle  is   true   for  division. 

But,  when  the  two  members  of  an  inequality  are  multiplied  oi 
divided  hy  a  negative  number,  the  inequality  will  subsist  in  a  con 
trary  sense. 

Take,  for  example,    8  >  7  ;    multiplying  by   —  3,   we  have 

—  24  <  —  21. 

Q                      R                 7 
In  like  manner,    8  >  7   gives    — — ,    or —  < — . 

Therefore,  when  the  two  members  of  an  inequality  are  multi- 
plied or  divided  by  a  number  expressed  algebraically,  it  is  ne- 
cessary to  ascertain  whether  the  multiplier  or  divisor  is  negative ; 
for,  in  that  case,  the  inequality  will  exist  in  a  contrary  sense. 

4.  It  is  not  permitted  to  change  the  signs  of  the  two  members  of 
an  inequality,  unless  we  establish  the  resulting  inequality  in  a  con- 
trary  sense ;  for  this  transformation  is  evidently  the  same  as  mul- 
tiplying the  two  members  by   —  1. 

5.  Both  members  of  an  inequality  between  positive  numbers  can 
be   squared,   and   the   inequality  will  exist  in   the  same  sense. 

Thus,   from  5  >  3,   we   deduce,   25  >  9  ;    from    a  +  i  >  c,  Ave 

find 

(a  +  bf  >  c2. 


CHAP.  IV.]  OF    INEQUALITIES  107 

6.  When  the  signs  of  both  members  of  the  irieqiiality  are  not 
hiown,  we  cannot  tell  hfore  the  operation  is  performed,  in  which 
sense  the  resulting  inequalitij  will  exist. 

For  example,  —  2<3  gives  (—2)2  or  4<9;  but  3>— 5 
gives,  on  the  contrary,    (3)^   or    9  <  (—  5)^    or  25. 

We  must,  then,  before  squaring,  ascertain  the  signs  of  the  tv.o 
members. 

EXAMPLES. 

1.  Find  the  limit  of  the  value  of  x  in  the  expression 

5a;  —  6  >  19.  Ans.  a;  >  5 

2.  Find  the  limit  of  the  value  of  x  in  the  expression 

14 
3a;  H a;  —  30  >  10.  Ans.  a?  >  4. 

3.  Find  the  limit  of  the  value   of  x  in  the  expression 

1  1  a;         13        17  , 

-—  X X  -\ f >  — .       Ans.  a;  >  o. 

6  322'^2  '^ 

4.  Find  the  limit  of  the  value  of  x  in  the  inequalities 

era;  d^ 

y  —  oa;  +  «6  <  y. 

5.  The  double  of  a  number  diminished  by  5  is  greater  than 
25,  and  triple  the  number  diminished  by  7,  is  less  than  double 
the  number  increased  by  13.  Required  a  number  which  shall 
satisfy  the  conditions. 

By  the  question,  we  have 

2a;  —  5  >  25. 

3a;  —  7  <  2a;  +  13. 

Resolving  these  inequalities,  we  have  a;  >  15  and  x  <  20. 
Any  number,  therefore,  either  entire  or  fractional,  comprised  be- 
tween  15  and  20,  will  satisfy  the  conditions. 


JOS  ELEMENTS    OF    ALGEBRA.  [CHAP.    * 


CHAPTER  V. 

EXTRACTION  OF  THE  SQUARE  ROOT  OF  NUMBERS. FORMATION  OF 

THE  SQUARE  AND  EXTRACTION  OF  THE  SQUARE  ROOT  OF  ALGE- 
BRAIC QUANTITIES. CALCULUS  OF  RADICALS  OF  THE  SECOND 

DEGREE. 

116.  The  square  or  second  power  of  a  number,  is  the  product 
which  arises  from  muhiplying  that  number  by  itself  once  :  for 
example,  49  is   the   square   of  7,  and   144   is  the  square   of  12. 

The  square  root  of  a  number,  is  that  number  which  multiplied 
by  itself  once  will  produce  the  given  number.  Thus,  7  is  the 
square  root  of  49,  and  12  the  square  root  of  144  :  for,  7x7  =  49, 
and    12  X  12  =  144. 

The   square  of  a  number,   either    entire   or  fractional,  is    easily 
found,  being  always  obtained  by  multiplying  the  number  by  itself 
once.     The    extraction   of  the    square    root   is,  however,   attended 
with  some  difficulty,   and  requires  particular  explanation. 
The  first  ten  numbers  are, 

1,     2,     3,       4,       5,       6,       7,       8,       9,       10, 
and  their  squares, 

1,     4,     9,     16,     25,     36,     49,     64,     81,     100: 
'and  reciprocally,  the  numbers  of  the  first  line  are  the  square  roots 
of  the  corresponding  numbers   of  the   second.     We  may  also  re- 
mark that,  the  square  of  a  number  expressed  by  a  single  figure,  will 
contain  no  figure  of  a  higher  denomination  than  tens. 

The  numbers  of  the  last  line,  1,  4,  9,  16,  &c.,  and  all  other 
numbers  which  can  be  produced  by  the  multiplication  of  a  num- 
ber by  itself,  are  called  perfect  squares. 

It  is  obvious,  that  there  are  but  nine  perfect  squares  among  all 
the  numbers  which  can  be  expressed  by  one  or  two  figures  :  the 
square  roots  of  all  other  numbers  expressed  by  one  or  two  figures 


CHAP,  v.]    EXTRACTION  OF  THE  SQUARE  ROOT  OF  NUMBERS.  109 

will  be  found  betv/een  two  whole  numbers  differing  from  each 
other  by  unity.  Thus,  the  square  root  of  55,  comprised  between 
the  perfect  squares  49  and  64,  is  greater  than  7  and  less  than  8. 
Also,  the  square  root  of  91,  comprised  between  the  perfect  squares 
81  and   100,  is  greater  than  9  and  less   than  10. 

Every  number  may  be  regarded  as  made  up  of  a  certain  num- 
ber of  tens  and  a  certain  number  of  units.  Thus  64  is  made  up 
of  6  tens    and   4    units,   and   may  be    expressed   under   the    form 

60  +  4  =  64. 
Now,  if  we  represent  the  tens   by  a   and  the  units  by  6,   we 
shall  have 

a-\-h     =60  +  4, 
and  (a  +  bf  =  (60  +  4)2, 

and  consequently, 

a2  +  2ab  +  b^    =  (60)2  _^  2  X  60  X  4  +  (4)2  =  4096. 
Hence,  the  square  of  a  number  composed  of  tens  and  units  con- 
tains, the  square  of  the  tens,  plus  twice   the  product  of  the  tens  by 
the  units,  plus  the  square  of  the  units. 

117.  If  now,  we  make  the  units  1,  2,  3,  4,  &c.,  tens,  by  an- 
nexing to   each     a  cipher,  we   shall  have, 

10,     20,     30,     40,       50,       60,       70,       80,       90,        100; 
and  for  their   squares, 

100,  400,  900,  1600,  2500,  3600,  4900,  6400,  8100,  10000, 
from  which  we  see  that  the  square  of  one  ten  is  1 00,  the  square 
of  two  tens,  400,  &c. :  and  hence,  the  square  of  tens  will  contain  no 
figure  of  a  less  denomination  than  hundreds,  nor  of  a  higher  name 
than   thousands. 

Let  us  now  take  any  number,  as  78,  and  square  it.     We  have 

78  z:^  70  +  8  ; 

that  is,  equal  to  7  tens,  or  70,  plus  8  units. 

Seven  tens,  or  70  squared     -         -  (70)2  _  4900 

twice  the  tens  by  the  units  is,     2  X  70  x  8  =  1120 

square  of  the  units  is,    -         -         -  (8)^  =      64 

lience, (78)2  —  G084. 

Let  us  now  reverse  this  process  anc?  ftad  the  square  root  of 
6084. 


110 


ELEMENTS    OF    ALGEBRA. 


[CHAP.  V 


60  84  1 78 
49 

1184 
1184 


Since   this    number  is   composed  of  more    than   two 
places  of  figures,  its  roots  will  contain  more  than  one.  60  84 

But  since   it  is  less  than   10000,  which  is  the  square 
of  100,  the  root  will  contain  but  two  figures  ;    that  is,  units  and 
tens. 

Now,  the  square  of  the  tens  must  be  found  in  the  two  left- 
hand  figures  which  we  will  separate  from  the  other  two,  by 
placing  a  point  over  the  the  place  of  units,  and  another  over 
the  place  of  hundreds.  These  parts,  of  two  figures  each,  are 
called  periods.  The  part  60  is  comprised  between  the  two  squares 
49  and  64,  of  which  the  roots  are  7  and  8  :  hence,  7  is  the  figure 
of  the  tens  sought ;  and  the  required  root  is  composed  of  7  tens 
and   a   certain  number  of  units. 

The  figure  7  being  found,  we 
write  it  on  the  right  of  the  given 
number,  from  which  we   separate 

it   by   a   vertical   line  :    then   we  7x2  =  148 

subtract   its    square    49    from    60, 

which  leaves   a  remainder  of  11,  0 

to  which  we  bring  down  the  two 
next  figures  84.  The  result  of  this  operation  is  1184,  and  this 
number  is  made  up  of  twice  the  product  of  the  tens  by  the  units 
plus   the   square   of  the   units. 

But  since  tens  multiplied  by  units  cannot  give  a  product  of  a 
less  name  than  tens,  it  follows  that  the  last  figure  4  can  form  no 
part  of  the  double  product  of  the  tens  by  the  units  :  this  double 
product  is  therefore  found   in   the   part   118. 

Nov/,  if  we  double  the  tens,  which  gives  14,  and  then  divide 
118  by  14,  the  quotient  8  is  the  units'  figure  of  the  root,  or  a 
figure  greater  than  the  units'  figure.  This  quotient  figure  can 
never  be  too  small,  since  the  part  118  will  be  at  least  equal  to 
twice  the  product  of  the  tens  by  the  units :  but  it  may  be  too  large ; 
for,  the  118  besides  the  double  product  of  the  tens  by  the  units, 
may  likewise   contain  tens   arising  from  the   square   of  the  units. 

To  ascertain  if  the  quotient  8  expresses  the  units,  we  write  the 
8  to  the  right  of  the  14,  which  gives  148,  and  then  Ave  multiply 
148  by  8.  Thus,  we  evidently  form,  1st,  the  square  of  tlie  units, 
and  2d,  the  double  product  of  the  tens  by  the  units.  This  mul- 
tiplication being  effected,  gives  for  a  product  USl,  a  number  equal 


CIIVP.  v.]    EXTRACTION  OF  THE  SQUARE  ROOT  OF  NUMBERS.  Ill 

to  the  result  of  the  first  operation.  Having  subtracted  the  prod- 
uct, Ave  find  the  remainder  equal  to  0  :  hence  78  is  the  root 
required. 

Indeed,  in  the  operations,  we  have  merely  subtracted  from  the 
given  number  6084,  1st,  the  square  of  7  tens  or  of  70;  2d,  twice 
ihe  product  of  70  by  8  ;  and  3d,  the  square  of  8  :  that  is,  the 
ihree  parts  which  enter  into  the  composition  of  the  square  of  78. 

Remark. — The  operations  in  the  last  example  have  been  per- 
formed on  but  two  periods.  It  is  plain,  however,  that  the  same 
reasoning  is  equally  applicable  to  larger  numbers  ;  for,  by  chan- 
ging the  order  of  the  units,  we  do  not  change  the  relation  in 
which  they  stand  to   each  other. 

Thus,  in  the  number  GO  84  95,  the  two  periods  60  84,  have 
the  same  relation  to  each  other,  as  in  the  number  6084  ;  and 
hence,  the  methods  pursued  in  the  last  example  are  equally  ap- 
plicable  to  larger  numbers. 

Hence,  for  the  extraction  of  the  square  root  of  numbers,  we 
have  the  following 

RULE. 

I.  Separate  the  given  number  into  periods  of  two  figures  each,  be- 
ginning at  the  right  hand :  the  period  on  the  left  will  often  contain 
but  one  figure. 

II.  Find  the  greatest  square  in  the  first  period  on  the  left,  and 
place  its  root  on  the  right  after  the  manner  of  a  quotient  in  division. 
Subtract  the  square  of  the  root  from  the  first  period,  and  to  ihe 
remainder  bring  down  the  second  period  for  a  dividend. 

III.  Double  the  root  already  found  and  place  it  on  the  left  for  a 
divisor.  Seek  how  many  times  the  divisor  is  contained  in  the  divi- 
dend, exclusive  of  the  right-hand  figure,  and  place  the  figure  in  the 
root  and  also  at  the  right  of  the  divisor. 

IV.  Multiply  the  divisor  thus  augmented,  by  the  last  figure  of 
the  root,  and  subtract  the  product  from  the  dividend,  and  to  the  re- 
mainder bring  down  the  next  period  for  a  new  divide7id. 

V.  Double  the  whole  root  already  found,  for  a  new  divisor,  and 
continue  the  operation  as  before,  until  all  the  periods  are  brought 
down. 

I.  Remark. — If,  after  all  the  periods  are  brought  down,  there  is 
no  remainder,  the  proposed  number  is  a  perfect  square.     But   if 


112  ELEMENTS    OF    ALGEBRA.  [CHAP.   V 

there  is  a  remainder,  we  have  only  found  the  root  of  the  greatest 
perfect  square  contained  in  the  given  number,  or  the  entire  part 
of  the  root  sought. 

For  example,  if  it  were  required  to  extract  the  square  root  of 
168,  we  should  find  12  for  the  entire  part  of  the  root  and  a  re- 
mainder of  24,  which  shows  that  168  is  not  a  perfect  square. 
But  is  the  square  of  12  the  greatest  perfect  square  contained  in 
168?  That  is,  is  12  the  entire  part  of  the  root?  To  prove  this, 
we  will  first  show  that,  the  difference  between  the  squares  of  two 
consecutive  numbers,  is  equal  to  twice  the  less  number  augmented 
by  unity. 

Let  a     =     the  less  number, 

and  a  -f  1     =     the  greater. 

Then  (a  +  1)2  =  a^  +  2a  +  1 

and  (a)2  =  a^ 

Their  difference  is         =  2a  +  1     as   enunciated. 


Hence,  the  entire  part  of  the  root  cannot  be  augmented  by  1,  «»• 
less  the  remainder  is  equal  to,  or  exceeds  twice  the  root  found,  plus 
unity. 

But,  12  X  2  +  1  =  25  ;  and  since  the  remainder  24  is  less 
than  25,  it  follows  that  12  cannot  be  augmented  by  a  number  as 
great  as  unity :    hence,  it  is  the   entire   part  of  the  root. 

The  principle  demonstrated  above,  may  be  readily  applied  in 
finding  the   squares   of  consecutive  numbers. 

If  the  numbers  are  large,  it  will  be  much  easier  to  apply  the 
above  principle  than  to  square  the   numbers   separately. 

For  example,  if  we   have     (651)2  =  423801; 
and  wish  to  find  the   square   of  652,  we   have 
(651)2  ^  423801 
+  2  X  651     =:       1302 

+  1     = l_ 

and  (652)2  =  425104. 

Also,  (652)2  =  425104 

+  2  X  652     =       1304 

+  1     =  1 


(653)2  _  426109. 


CHAP,   v.]    EXTRACTION"  OF  THE  SQUARE   ROOT  OF  NUMBERS.  113 

II.  Remark. — The  number  of  figures  in  the  root  will  always 
be  equal  to  the  number  of  periods  into  which  the  given  number 
is  separated. 

examples. 

1.  To  find  the  square  root  of  7225. 

2.  To  find  the  square  root  of  17689. 

3.  To  find  the  square  root  of  994009. 

4.  To  find  the  square  root  of  85678973. 

5.  To  find  the  square  root  of  67812675. 

Of  Incommensurahle  Nvmhers. 

118.  If  a  number  is  not  a  perfect  square,  its  square  root  is 
said  to  be  incommensurahle,  or  irrational,  because  it  cannot  be  ex- 
pressed in  terms  of  the  numerical  imit.  Thus,  y  2  ,  v  5  ,  v  7  , 
are  incommensurable  numbers.  They  are  also  sometimes  called 
radicals  or  surds. 

Two  or  more  numbers  are  sa'-t  ic  oe  pr.n'.  wi'a  ie'"p'"jT  v 
each  other,  when  there  is  no  whole  number  except  unity  which 
will  divide  each  of  them  without  a  remainder.  Thus,  the  num- 
bers 3  and  5  are  prime  with  respect  to  each  other  ;  and  so  also 
are  4   and   7   and  9. 

In  order  to  prove  that  the  root  of  an  imperfect  power  cannot 
be   expressed  by  exact  parts   of  unity,  we   must  first  show  that, 

Every  numher  P,  ichich  ivill  exactly  divide  the  product  A  X  B 
of  two  numbers,  and  which  is  prime  with  one  of  them,  will  divide 
the  other. 

Let  us  suppose  that  P  will  not  divide  A,  and  that  A  is  greater 
than  P. 

Let  us  now  find  the  greatest  common  divisor  of  A  and  P.     If 
we   represent   the    entire   quotients    by   Q,   Q',   Q''^,   he,  and  the 
remainders,  respectively,  by  R,  R',  R^^,  &c. ;   we  shall  have 
A    ||P_ 

Q  ' 

P   i|_R_ 

Q" 
R  IIJR^ 

R',|R- 


hence, 

A  =  PQ  +  R, 

hence, 

P  =  RQ'  +  R', 

hence. 

R  =  R'Q'^  +  R", 

hence. 

R'=  \V'Q."'-\-  R'" 

114  ELEMENTS    OF    ALGEBRA.  [CHAP.   V. 

Now,  since  the  remainders  R,  R^,  R^^  &c.,  constantly  dimin- 
ish, if  the  division  be  continued  sufficiently  far,  we  shall  obtain 
a  remainder  equal  to  unity ;  for  the  remainder  cannot  be  0,  since 
by  hypothesis  A  and  P  are  prime  with  each  other.  Hence,  we 
have    the  following  equations  : 

A  =  P  Q     +R 
P   =  R  Q'    +  R' 
R  =  R'  Q'^  +  R'' 
R^  =  R'^Q'^'  +  W 


Multiplying  the  first  of  these  equations  by  B,  and  dividing  by 
P,  we  have 


But,  by  hypothesis,    — — —    is  an   entire   number,  and  since  B 

and  Q  are  entire  numbers,  the  product  BQ  is  an  entire  number. 

Hence,  it  follows  that       ^        is  an  entire  number. 

If  we  multiply  the  second  of  the  above  equations  by  B,  and 
divide  by  P,  we  have 

^       BRQ^   ,    BR^ 

But  we  have  already  shown,  that    — -—    is  an  entire  number ; 

BRQ'    .  .  u         rr^x,.    u  •        1.  BR' 

hence    — — —    is  an  entire  number.      Ihis  being  the  case,  - 

must  also  be  an  entire  number.  If  the  operation  be  continued 
until  the  number  which   multiplies  B  becomes  1,  we   shall   have 

B  X  1 

— :g—  equal  to  an  entire  number,  which  proves  that  P  will  di- 
vide B. 

In  the  operations  above  we  have  supposed  A  >  P ;  but  if 
P  >  A,  we  should  first  divide  P  by  A. 

Hence,  if  a  number  P  will  exactly  divide  the  product  of  two  num- 
bers, and  is  prime  with  one  of  them,  it  will  divide  the  other 


CHAP,  v.]  EXTRACTION  OF  THE  SQUARE  ROOT  OF  FRACTIONS.     115 

We  see  from  what  has  preceded  that,  if  P  is  prime  with  re- 
spect to  any  number  as  a,  it  will  also  he  prime  with  respect  to  d? 
and  the  higher  powers  of  a. 

For,  if  P  will  divide  a^  =  a  X  a,  it  must  divide  one  of  the  fac- 
tors a  or  a.  But  this  would  be  contrary  to  the  supposition ;  hence, 
P  cannot  divide  a^.  In  the  same  way  it  may  be  proved  that  it 
cannot  divide  the  higher  powers   of  a. 

We  will  now  show  that  the  square  root  of  an  imperfect  square 
cannot  be  expressed  by  a  fractional  nmnber. 

Let  c  be  an  imperfect  square.  Then  if  its  exact  root  can  be 
expressed  by  a  fractional  number,  we  can  assume 

in  which   the   fraction    —r-    is   in   its   lowest  terms :    that   is,   in 

0 

which  a  and  b  are   prime  with  respect  to  each  other. 

Now,  if  we  square  both   members  of  the  equation,  Ave  have 

'  =  ¥^ 

in  which  c  is  an  entire  number  ;    and  hence,  if  the  equation  is 
true,  a^  must  be  divisible  by  h"^. 

But  if  a^  is  divisible  by  6^,  the  product  a  x  a  ^^  a^,  must  be 
divisible  by  b ;  for  the  division  would  be  effected  by  dividing 
twice  by  b.  But  we  have  seen  that  a^  is  not  divisible  by  h ; 
therefore,  we  cannot  express  the  square  root  of  an  imperfect 
square  by  a  fractional  number. 


Extraction  of  the  Square  Root  of  Fractions. 

119.  Since  the  second  power  of  a  fraction  is  obtained  by  squar- 
ing the  munerator  and  denominator  separately,  it  follows  that  the 
square  root  of  a  fraction  will  be  equal  to  the  square  root  of  the 
numerator  divided  by  the  square  root  of  the  denominator. 

a 


For  example,  V  t?  =  -t-» 

▼    6''         0 


a  a         a' 


116  ELEMENTS    OF    ALGEBRA.  [CHAP.  V. 

But  if  the  numerator  and  the  denominator  are  not  both  perfect 
squares,  the  root  of  the  fraction  cannot  be  exactly  found.  We 
can,  however,  easily  find  the  exact  root  to  within  less  than  one 
of  the  equal  parts  of  the  fraction.     For  this  purpose, 

Multiphj  both  terms  of  the  fraction  by  the  denominator — this  makes 
the  denominator  a  perfect  square.  Then  extract  the  square  root  of 
the  perfect  square  nearest  the  value  of  the  numerator,  and  place  the 
root  of  the  denominator  under  it — this  fraction  will  be  the  approxi- 
mate root. 

3 

Thus,  if  it   be  required  to  extract  the   square  root  of    — ,    we 

15 
multiply  both   terms  by   5,   which  gives    — :    the   square  nearest 

4  .  .  . 

15  is  16:  hence,    —    is  the  required  root,  and  is  exact  to  with- 
5 

in  less  than    -— . 
5 

120.  We  may,  by  a  similar  method,  determine,  approximatively, 
the  roots  of  whole  numbers  which  are  not  perfect  squares.  Let 
it  be   required,  for  example,   to   determine    the    square  root  of  an 

entire   number  a,  nearer  than  the   fraction    —  ;    that  is  to  say,  to 

n 

find  a  number  which  shall  differ  from  the  exact  root  of  a,  by  a 
quantity  less  than    — .     It  may  be  observed  that, 


an 


o  =  — 

n 


If  we    designate   by  r   the    entire  part  of  the   root  of  on',  the 
number  an^  will  then  be  comprised  between  r^  and  (r  +  1  )2 ;  and 

will  be  comprised  between    — -     and    ^ ~- ;    and  conse 

n^  n''  n^ 

quently  the  true  root  of  a  is  comprised  between 


r!  and    ^/^}l. 

r  r  +  1 

that   is,  between    —    and    .      But   the  diflerence   between 

n  n 

1  r 
these  numbers  is    — :    hence    —    will  represent  the  square  root 

n  n 


CHAP,   v.]    EXTRACTION  OF  THE   SQUARE  ROOT  OF  FRACTIONS.  117 

of  a  Avithin  less  than  the  fraction    — .     Hence  to  obtain  the  root : 

n 

Multiply  the  given  number  by  the  square  of  the  denoj/iinator  of 
the  fraction  which  determines  the  degree  of  approximation  :  then  ex- 
tract the  square  root  of  the  product  to  the  nearest  unit,  and  divide 
this  root  by  the  denominator  of  the  fraction. 

1.  Suppose,  for  example,  it  were  required  to  extract  the  square 

root  of  59,  to  within  less  than    — . 

First,  (12)"  =  144;    and    144  X  59  =  8496. 

Now,  the  square  root  of  8496  to  the  nearest  unit,  is  92  :  hence 

92  1 

—  =  7  A,    which  is  true  to  within  less  than    — . 
12  ^-'  12 


2    To  find  the  vll  to  within  less  than    — . 
^  15 


3    To  find  the  -v/223  to  within  less  than    — . 
^  40 


Ans.  3 — -. 
15 


37 

Ans.   14—-. 

40 


121.  The  manner  of  determining  the  approximate  root  in  deci- 
mals, is  a  consequence  of  the  preceding  rule. 

To    obtain   the    square    root   of   an    entire    number    within     — , 

-— -,  ,  &c.,  it  is  only  necessary  according  to  the  preceding 

rule,  to  multiply  the    proposed  number  by  (10)^,   (100)^,  (lOOO)^; 
or,  which  is  the  same  thing, 

Annex  to  the  number,  two,  four,  six,  (Sf-c,  ciphers :  then  extract 
the  root  of  the  product  to  the  nearest  unit,  and  divide  this  root  hy 
10,  100,  1000,  &c.,  which  is  effected  by  pointing  off  one,  two,  three, 
iSfC,  decimal  places  from  the  right  hand. 

EXAMPLES. 


1.  To  find  the  square  root  of  7  to  within    


100' 


118 


ELEMENTS    OF    ALGEBRA. 


Having  multiplied  by  (100)^,  that  is, 
having  annexed  four  ciphers  to  the  right 
hand  of  7,  it  becomes  70000,  whose 
root  extracted  to  the  nearest  unit,  is 
264,  which  being  divided  by  100  gives 
2.64  for  the  answer,  which  is  true  to 
1 

Too* 

1 
100' 
1 


within  less  than 

2.  Find  the  ^/29  to  within 

3.  Find  the  >/ 227    to  within 


70000 
4 


[CHAP.  V. 

2.64 


46 


524 


300 
276 


2400 
2096 


10000 


304  Rem. 
Ans.  5.38. 
Ans.  15.0665. 


Remark. — The  number  of  ciphers  to  be  annexed  to  the  whole 
number,  is  always  double  the  number  of  decimal  places  required 
to  be  found  in  the  root. 

122.  The  manner  of  extracting  the  square  root  of  decimal  frac- 
tions is  deduced  immediately  from  the  preceding  article. 

Let  us  take   for  example   the   number  3.425.     This   fraction  is 

3425 
equivalent  to    .     Now   1000  is  not  a  perfect  square,  but  the 

denominator  may  be  made  such  without  altering  the  value  of  the 

,      ,  1        r.       I.-       •  34250 

fraction,  by  multiplymg  both   the  terms  by  10;    this  gives 


or 


34250 

(ioo)5- 


Then   extracting   the    square   root   of    34250    to    the 
nearest  unit,  we   find  185  ;    hence    — -    or  1.85  is  the  required 

root  to  within    — — . 

If  greater  exactness  be  required,  it  will  be  necessary  to  add  to 
the  number  3.4250  so  many  ciphers  as  shall  make  the  periods 
of  decimals  equal  to  the  number  of  decimal  places  to  be  found  in 
the  root.     Hence,  to  extract  the  square  root  of  a  decimal  fraction : 

Annex  ciphers  to  the  proposed  number  until  the  number  of  deci- 
mal places  shall  he  equal  to  double  the  number  required  in  the  root. 
Then  extract  the  root  to  the  nearest  unit,  and  point  off  from  tht 
right  hand  the  required  number  of  decimal  places. 


CHAP,  v.]    SQUARE  ROOT  OF  ALGEBRAIC  QUANTITIES.  119 

EXAMPLES. 

1.  Find  the     -/ 3271.4707    to  within    .01.  Ans.  57.19. 

2.  Find  the     -/ 31.027    to  within    .01.  A71S.  5.57. 

3.  Find  the     -/ 0.01001    to  within    .00001.        Ans.  0.10004. 

123.  Finally,  if  it  be  required  to  find  the  square  root  of  a  vul 
gar  fraction  in  terms  of  decimals  : 

Change  the  vulgar  fraction  into  a  decimal  and  continue  the  di- 
vision until  the  number  of  decimal  places  is  double  the  number  re- 
quired in  the  root.  Then  extract  the  root  of  the  decimal  by  the  last 
rule. 

EXAMPLES. 

1.  Extract  the    square  root  of    —    to  within  .001.     This   nuni 

Der,  reduced  to    decimals,  is    0.785714  to  within  0.000001.     The 
root  of  0.785714  to  the  nearest  unit,  is  .886 :   hence  0.886  is  the 

root  of    —    to  witliin  .001. 
14 


\/  2- 


13 
2.  Find  the   \/  2—    to  within  0.0001.  Ans.  1.6931. 

15 


Extraction  of  the  Square  Root  of  Algebraic   Quantities. 

124.  Let  us  first  consider  the  case  of  a  monomial.  In  order 
to  discover  the  process  for  extracting  the  square  root,  let  us  see 
how  the  square  of  a  monomial  is  formed. 

By  the  rule  for  the  multiplication  of  monomials  (Art.  41),  we 
have 

(5^253^)2  _  5^253^  X  5a^^c  =  25a^^c^ ; 

that  is,  in  order  to  square  a  monomial,  it  is  necessary  to  .square 
its  co-eficient,  and  double  the  exponent  of  each  letter.  Hence,  to 
find  the  square  root  of  a  monomial, 

1st.  Extract  the  square  root  of  the  co-eficicnt  and  divide  the  ex- 
ponent of  each  letter  bi/  two.  2d.  To  the  root  of  the  co-effLcievf 
annex  each  letter  with  its  new  exponent,  and  the  result  will  he  tli/: 
required  root. 


120  ELEMENTS    OF    ALGEBRA.  [CHAP.   V 

Thus,     -/64a6^/4  =:  8a^^  ;    for,    80^^*2  ^  q^W  —  Gia'^b^ 
and,         -y/eSSa-'iV  _  25ab^c^  ;     for,    {25ab*c^y  =  625a^h<^. 

125.  From  the  preceding  rule,  it  follows,  that,  when  a  monomial 
is  a  perfect  square,  its  numerical  co-efficient  is  a  perfect  square,  and 
the  exponent  of  every  letter  an  even  number.  Thus,  25a'*i2  is  a 
perfect  square,  but  98ai*  is  not  a  perfect  square ;  for,  98  is  not 
a  perfect  square,  and  a  is  affected  with  an  uneven  exponent. 

An  imperfect  square  is  introduced  into  the  calculus  by  affecting 
it  with  the  radical  sign  y  ,  and  written  thus,  y  98a6*.     Quan- 

tities of  this  kind  are  called  radical  quantities,  or  irrational  quan- 
tities, or  simply  radicals  of  the  second  degree. 

These  expressions  may  sometimes  be  simplified.  For,  by  the 
definition  of  the   square   root,  we  have 

^/a'    ^X  y/ a        =  {-y/ af     =  a, 
yob        X  y  ab       ^[yuby      =z  ab, 
yabc      X  -y/abc     =  [yahc)'^    =  abc, 
■y  abed    X  y  abed  =  (yabcd)'^  =  abed; 
and  the  same  would  be  true  for  any  number  of  factors. 
Again, 

{-/7.  /T .  /7.  /7)2  =  (77)2 .  (^^Yy. .  (y7')2 ,  (/T)2  ^  abed, 

by  the  rule  for  multiplying  monomials  (Art.  41). 

Now,  since,  [yabcdy  =^  abed, 

and,        (y   a.yb.yc.-\/dy  =  abed ; 
it  follows,  that  the  quantities  themselves  are   equal :  hence, 
yf  abed  z=:  yj  a  .  yj  b  .  y  c  .  y  d  ;    that  is, 

The  square   root  of  the  product  of  two  or  more  factors  is  equal  to 
the  product   of  the   square  roots   of  those  factors. 
This  being  proved,  we  can  write 

y98ai*  —  ^496*  X  2a  -  ^49^  X  y/2a. 
But,  yisT^  -  7^2  . 

hence,  -/98ai^  =  7^2  y/2a. 


CHAP.  V.J     SQUARE  ROOT  OF  ALGEBRAIC  QUANTITIES.  121 

In  like  manner, 

^A5a-Pc^d  =  yf^aWc^  X  ^hd  =  3abc  y/dbd. 
■^864a"b^c^^  =  ^/ I4iu^h*c^^  X  6bc  —  \2ab\^  -/oic. 

The  quantity  which  stands  Avithout  the  radical  sign  is  called 
the  co-cjicicnt  of  the  radical.  Thus,  7Z/-,  3ahc,  and  12ub"c^,  are 
co-efflcienls  of  the  radicals. 

In  general,  to  simplify  a  radical  of  the  second  degree : 

Divide  the  quantity  under  the  radical  sign  by  the  smallest  mono- 
mial, with  reference  to  its  co-eff,cients  and  exponents,  that  will  give 
for  a  quotient  a  perfect  square.  Then,  extract  the  root  of  the  per- 
fect square  and  place  it  without  the  radical  sign,  under  which,  write 
the  monomial  used  as  a  divisor. 

EXAMPLES. 

1.  To  reduce  ■y/loa^bc    to  its  simplest  form. 

2.  To  reduce  y  128i^a^J2    ^q  ^^g  simplest  form. 

3.  To  reduce  y  32a^6^c    to  its  simplest  form. 

4.  To  reduce  -v/256a26^c^    to  its  simplest  form. 

5.  To  reduce  -s/ lQ2Aa%'' c^     to  its  simplest  form. 

6.  To  reduce  ■y/728a''b^c^d     to  its  simplest  form. 

126.  Since  like  signs  in  both  the  factors  give  a  plus  s-ign  m 
the  product,  the  square  of  —  a,  as  well  as  that  of  +  a,  will  be 
a^ :  hence,  the  root  of  a^  is  either  -\-  a  or  —a.  Also,  the  square 
root  of  25a2i*  is  either  +  5ab^  or  —  dab"^.  Whence  we  may  con- 
clude, that  if  a  monomial  is  positive,  its  square  root  may  be  af 
fected  either  with  the  sign  +   or   —  ; 

thus,  -/9a*  =  ±  3a2, 

for,    +  Sa^    or    —  3a^,    squared,  gives    9a*.     The  double    sign    ± 
with  which  the   root  is   affected,  is  read  plus  or  mimls. 

If  the  proposed  monomial  were  negative,  it  would  have  no  square 
root,  since  it  has  just  been  shown  that  the  square  of  every  quan- 
tity, whether  positive  or  negative,  is  essentially  positive.  There- 
fore, 

V—  9,     -/  —  4a2,     ^/—  Qd^b, 
are  algebraic    symbols    which   indicate    operations   that    cannot   be 


122  ELEMENTS    OF    ALGEBRA.  fCHAP.  V. 

performed.  They  are  called  imaginary  quantities,  or  rather,  im 
aginary  expressions,  and  are  frequently  met  with  in  the  resolution 
of  equations  of  the  second  degree. 

127.  Let  us  now  examine  the  law  of  formation  of  the  square 
of  a  polynomial ;  for,  from  this  law,  the  rule  is  deduced  for  ex- 
tracting  the  square  root. 

It  has  already  been  shown  (Art.  46),  that, 

(a  -I-  If  =  a2  +  2ah  +  IP- ;    that  is, 

The  square  of  a  Unomial  is  equal  to  the  square  of  the  first  term 
plus  twice  the  product  of  the  first  term  by  the  second,  plus  the 
square  of  the  second. 

The  square  of  a  pol}Tiomial,  is  the  product  arising  from  midti- 
plying  the  polynomial  by  itself  once  :  hence,  the  first  term  of  the 
product,  arranged  with  reference  to  a  particular  letter,  is  the 
square  of  the  first  term  of  the  polynomial  arranged  with  reference 
to  the  same  letter.  Therefore,  the  square  root  of  the  first  terra 
of  such  a  product  will  be  the  first  term  of  the  required  root. 

128.  Let  us  now  extract  the  square  root  of  the  poljmomial 

28^5  -j-  49a*  +  4a6  +  9  +  42^2  +  12a3 , 
which  arranged  with  reference  to  the  letter  a,  becomes, 


4a6  +  28a5  +  49a4  +  12a3  +  42a-  +  9 

2a3+    7a2  +  3 

4o6 

4a3+    7a2 

R    =28a5  +  49a*+12a3-i-42a2  +  9 

7a2 

28c5  +  49a* 

28a5+49a*  =  (2r4-r')/ 

R'  =         -         -          12a3  4-42a2  +  9 
12a3  +  42a2  +  9 

4a3+14a2+3 
3 

W=        -         -            0           0         0 

12a2+  42a2-|-  9  =  (2»  +  r^y 

Now,  since  the  square  root  of  4a^  is  2a'^,  it  follows  that  2a^ 
is  the  first  term  of  the  required  root.  Designate  this  term  by  r, 
and  the  following  terms  of  the  root,  arranged  with  reference  to  a, 
by  /,  7^^,  r'^^,  &c. 

Now,  if  we  denote  the  given  polynomial  by  N,  we  shall  have 
N  =  (r  +  r'  +  r'^+  r"''  +  &c.  ;)2 
or,  if  we  designate  all  the  terms  of  the  root,  after  the  first,  by  i 
N  =  (r  +  5)2  =  7-2  +  2rs  +  ^2 

=  r2  +  2r  (t^  +  r^'  +  r"''  +  &c.)  +  ^2. 


CHAP.  V.l    SQUARE  ROOT  OF  ALGEURAIC  QUANTITIES.  123 

If  now  we  subtract  r^  =  4a^,  from  N,  and  designate  the  re- 
mainder by  R,  we  shall  have 

R  -  N  -  4a6  =  2r  (r'  +  r'^  +  r^''  +  &c.  )  +  *^ ; 

in  which  the  first  term  2rr^  will  contain  a  to  a  higher  power 
than  either  of  the  following  terms.  Hence,  if  the  first  term  of 
the  first  remainder  be  divided  by  twice  the  first  term  of  the  root, 
the  quotient  will  be  the  second  term  of  the  root. 

If  now,  we  place  r  -{-  r'  =z  n 

and  designate  the  remaining  terms  of  the  root,  r^',  r"\  &c.,  by  s^y 
we  shall  have 

N  =  («  +  sy  -  n2  +  2«/  +  ^2  ;    and 
R'  =  N  -  ra2  =  (2r  +  2/)  [r"  +  r"'  +  &c.)  +  ^'2 ; 

in  which,  if  we  perform  the  midtiplications  indicated  in  the  sec- 
ond member,  the  term  Irr"  will  contain  a  higher  power  of  a  than 
either  of  the  following  terms.  Hence,  if  the,  first  term  of  the 
second  remainder  be  divided  by  twice  the  first  term  of  the  root,  the 
quotient  will  he  the  third  term  of  the  root. 
If  we  make 

r  +  r^  +  r^"  =  ;/,    and    r'''  +  j-'^  +  &c.  =  ^', 
we  shall  have 

N  =  («'  +s"y  =  n'2  +  2ns"  +  s"^ ;    and 
R'^  =  N  -  n'3  =  2  (r  +  r^  +  r")  [f"  +  r'^  +  &c.)  +  5"^ ; 

from  which  we  see,  that  the  first  term  of  any  remainder,  divided 
by  twice  the  first  term  of  the  root,  will  give  a  new  term  of  the  re- 
quired  root. 

It  should  be  observed,  that  instead  of  subtracting  n"^  from  the 
given  polynomial,  in  order  to  find  the  second  remainder,  that  that 
remainder  could  be  found  by  subtracting  (2r  +  r')r'  from  the 
first  remainder.  So  the  third  remainder  may  be  found  by  sub- 
tracting (Jin  +  /^)  7^'  from  the  second,  and  similarly  for  the  re- 
mainders which  follow. 

In  the  example  above,  the  third  remainder  is  equal  to  zero, 
and  hence  the  given  polynomial  has  an  exact  root. 

Hence,  for  the  extraction  of  the  square  root  of  a  polynomial 
we  have  the  following 


124  ELEMENTS    OF    ALGEBRA.  [CHAP.  V 

RULE. 

I.  Arrange  the  polynomial  with  reference  to  one  of  its  letters^ 
and  then  extract  the  square  root  of  the  frst  term,  which  v:ill  givi 
the  frst  term  of  the  root.  Subtract  the  square  of  this  term  fro/r 
the  given  polynomial. 

II.  Divide  the  first  term  of  the  remainder  by  twice  the  first  term 
of  the  root,   and  the  quotient   will  be  the  second  term  of  the  root 

III.  From  the  first  remainder  subtract  the  product  of  twice  the 
first   term  of  the   root  phis  the  second  term,   by  the  second  term. 

IV.  Divide  the  first  term  of  the  second  remainder  by  twice  the 
first  term  of  the  root,  and  the  quotient  will  be  the  third  term  of 
the  root. 

V.  From  the  second  remainder  subtract  the  product  of  twice  the 
first  and  second  terms  of  the  root,  plus  the  third  term  by  the  third 
term,  and  the  result  will  be  the  third  remainder,  from  which  the 
fourth  term  of  the  root  may  be  found ;  and  proceed  in  a  similar 
manner  for  the  remaining  terms  of  the  root. 

EXAMPLES. 

1.  Extract  the  square  root  of  the  polynomial 

49aH^  —  24ab^  +  25a*  —  30a^  +  16&* 
First  arrange  it  with  reference  to  the  letter  a. 


25a*  —  30a^  -f  49a^^"  —  24ab^  +166* 
25a* 


R 

=  -  SOa^Z,  +  49a2i2  _ 
—  30a^  +     9a%^ 

-  24ab^  +  16^** 

R' 

=                  +  40a262  _ 
+  40a2i2  _ 

-  24ab^  +  16i* 
-24ab^  +  16M 

R''  = 


5a2  _  2ab  +  4b^ 
10a2  —  Sab 
—  3ab 

30^36"T9a^6^ 
10a2  —  Gab  +  4^2 
4i2 

40^2^2  :_  240^3  +  166* 


2.  Find  the  square  root  of 

a*  +  4a3a;  +  6a2a;2  +  4ax^  +  xK 

3.  Find  the  square  root  of 

a*  —  2a3a;  +  Sa^x"^  —  2ax^  +  x*. 

4.  Find  the  square  root  of 

4x6  _^  i2a:5  -f  5x*  —  2x^  +  7x^  —  2a:  +  1. 


CHAP,  v.]  SQUARE    ROOT    OF    ALGEBRAIC    QUANTITIES.  125 

5.  Find  the  square  root  of 

9a*  —  I2a^  +  28aW  —  16aP  +  IGb*. 

6.  Find  the  square  root  of 

25a^b^  —  AOa^^c  +  IGaWc^  —  \8a}P-c^  +  SGZ^V  —  30a*ic  +  2\a^hc^ 
—  36a2Jc3  +  9a4c2. 

129.  We  will  conclude  this  subject  with  the  following  remarks*. 

1st,  A  binomial  can  never  be  a  perfect  square.  For,  its  root 
cannot  be  a  monomial,  since  the  square  of  a  monomial  will  be  a 
monomial ;  nor  can  the  root  be  a  polynomial,  since  the  square  of 
the  simplest  polynomial,  viz.,  a  binomial,  will  contain  at  least 
three  terms.     Thus,  an  expression  of  the  form 

a2  -(-  J2 
can  never  be  a  perfect  square. 

2d.  A  trinomial,  however,  may  be  a  perfect  square.  If  so, 
when  arranged,  its  two  extreme  terms  must  be  squares,  and  the 
middle  term  double  the  product  of  the  square  roots  of  the  other 
two.  Therefore,  to  obtain  the  square  root  of  a  trinomial,  when  it 
is  a  perfect  square. 

Extract  the  roots  of  the  two  extreme  terms,  and  give  these  roots 
the  same  or  contrary  signs,  according  as  the  middle  term  is  posi- 
tive or  negative.  To  verify  it,  see  if  the  double  product  of  the  two 
roots  is  equal  to   the   middle   term   of  the  trinomial.     Thus, 

9a^  —  ASa^lj^  +  Q\a%^     is  a  perfect  square, 
for,         y^  =  3a3,      and      -/G4^  =  —  Sah"^, 
and  also,      2  x  Sa^  x  —  Sai^  —  _  48a'«i2,     the  middle  term. 
But  4a2  -f  I4ai  +  9^2 

is   not   a   perfect   square :    for,  although   Aa^   and    +  9^2   are   the 
squares  of  2a  and   2b,   yet  2  X  2a  X  3i    is  not  equal  to   14a J. 

3d.  When,  in  extracting  the  square  root  of  a  polynomial,  the 
first  term  of  any  one  of  the  remainders  is  not  exactly  divisible 
by  twice  the  first  term  of  the  root,  we  may  conclude  that  the 
proposed  polynomial  is  not  a  perfect  square.  This  is  an  evident 
consequence  of  the  course  of  reasoning,  from  which  the  general 
rule  for  extracting  the   square  mot  was  deduced. 


126  ELEMENTS    OF    ALGEBRA.  [CHAP.  V 

4th.  When  the  polynomial   is   not  a  perfect  square,  the   expres- 
sion for  its  square  root  may  sometimes  be   simplified. 
Take,  for  example,  the  expression 

-y/a^b  +  AaW  +  4aP. 
The  quantity  under  the  radical  is  not  a  perfect  square :  but  it 
can  be  put  under  the  form 

ab  (a2  +  4ab  +  4^2). 

Now,  the  factor  within  the  parenthesis  is  evidently  the  square  of 
a  +  2b,    whence  we  have 

■y/a^  +  4a^^  4-  4ab3  =  (a  +  2b)  yfab. 

Of  the  Calculus  of  Radicals  of  the  Second  Degree, 

130.  A  radical  quantity  is  the  indicated  root  of  an  imperfect 
power.  If  the  root  indicated  is  the  square  root,  the  expression  is 
called  a  radical  of  the  second  degree.     Thus, 

y/~^,      3  -/y,      7  ^/~2, 
are  radicals  of  the  second  degree. 

131.  Two  radicals  of  the  second  degree  are  similar,  when  the 
quantities  under  the  radical  sign  are  the  same  in  both.  Thus, 
3^/  b  and  5c  y  b  are  similar  radicals ;  and  so  also,  are  9  y  2 
and    7 -/Y^ 

Addition  and  Subtraction. 

132.  In  order  to  add  or  subtract  similar  radicals,  add  or  sub- 
tract their  co-eff,cients,  and  to  the  sum  or  difference  annex  the  com- 
mon radical. 

Thus,  3a  y/~b    +  5c  y/~b    —  (3a  +  5c)  V^; 

and  3a  y  b    —  5c  yb    =  (3c  —  5c)  y  b  . 

In  like  manner, 

7  V2a  +  3    V^  =  (7  +  3)  V^  =  10  V^'f 
and  7  V'2a  —  3    ^/2a  =  (7  —  3)  -/ia  =    4  ^/2a. 

Two  radicals,  which  do  not  appear  to  be  similar,  may  become 
so  by  simplification  (Art.   125). 


CUAP.  v.]  RADICALS    OF    THE    SECOXD    DEGREE.  127 

For  example, 

>/  iSai"'  -}   b  y/lba  =  \b  y/Va  +  bh  yfZa  =  9b  -/Sa  , 
and    2-/ 45       —  3y/~E~  =  6    -/T  —  3    y^  =  3   V~^ • 

When  the  radicals  are  not  similar,  their  addition  or  subtraction 
can  only  be  indicated.    Thus,  to  add  3  y  5    to   5  y  a ,  we  wTite 
5  ^/^  +  3  ^/T. 

Multiplication. 

133.  To  multiply  one  radical  by  another,  let  us  observe  that 

( V  c    X  y/by  =  ah  ;    also 
that,                     [s/aby  =  ab ;    hence, 

(V  a    X  \/  by  =z  [-s/aby  ;    and  consequently, 

■\/  a    X  y  b      =    yab :     that  is 

The  product  of  two  radicals  of  the  second  degree  is  equal  to  the 
square  root  of  the  product  of  the  quantities  under  the  radical  signs 

When  there  are  co-efRcients,  we  first  multiply  them  together,  and 
write  the  product  before  the  radical  sign.     Thus 

3  y/bab  X  4  ^/20a  =  12  y/\QOa^b  =  120a  -y/  b 
'     2a  yfbc  X  3a  yfhc  —  %a^  ^/¥^  =  6«^ic. 

2a  -/a2  +  52  X  —  3a  ^^^7^  —  _  Qq^  (a2  4.  52). 

Division. 

134.  To  divide  one  radical  by  another,  let  us  observe  that 

f^)'=i^  =  ^:    also, 

\y/b^     (Vby      b 

that  ( Y  -j-j    =  -f-  '•    hence 

y  a 


7=^  —  v  T"  •    t^3,t  is, 

■vA      ^  * 

T^e  quotient   of  two   radicals  is   equal   to  the   square   root  of  the 
quotient  of  the   quantities  under  the  radical  signs. 


128  ELEMENTS    OF    ALGEBRA.  [CHAP.    V. 

When  there   are   co-efficients,   write  their  quotient  as  the  co-efi- 
rient   of  the   radicals. 
For  example, 

5a     I  h 


5a  -y/T  ^  2b  -/T—  . 

And  I2ac  -y/Sbc-^  4c  y  2Z»  =  Say  — —  =  3a  y  3c. 

135.  There  are  two  transformations  of  frequent  use  in  fniJirig 
the  numerical  values  of  radicals. 

The  first  transformation  consists  in  passing  the  co-efficient  of 
a  radical  under  the  radical  sign.  Take,  for  example,  the  ex- 
pression 3a  y  5i.  By  applying  the  rules  for  the  multiplication  of 
radicals  we  may  write, 

3a  ^/5b  ^  ^/'^d^  X  >/5h  =  y/^a^  x  5h  =  y/~\5^. 
Therefore,  the  co-efficient  of  a  radical  may  be  passed  under  the  rai 
ical  sign,   as  a  factor,   by  squaring   it. 

The  principal  use  of  this  transformation,  is  to  find  an  approxi- 
mate value  of  any  radical,  which  shall  differ  from  its  true  value, 
by  less  than  unity. 

For  example,  take  the  expression  6y  13.  Now,  as  13  is  not 
a  perfect  square,  we  can  only  find  an  approximate  value  for  its 
square  root ;    and  when  this  approximate  value  is  multiplied  by  6, 

the  product  will  differ  materially  from  the  true  value  of  6  ylS- 
But  if  we  write, 

6  yii  =  -/62  X  13  =  -v/36  X  13  =  ^468, 
we   find  that  the    square   root  of  468    is    the    whole    number   21, 
plus  an  irrational  number  less   than  unity.     Hence, 

6  y  13  =  21     plus  an  irrational  number  less  than  1. 
In  a  similar  manner  we  may  find, 

12  y  7  =31     plus  an  irrational  number  less  than  1 

136.  Having  given  an  expression  of  the  form, 

a                           a 
or — =, 

P  +  V?  P  -  V  q 

in  which  a  and  p  are  any  numbers  whatever,  and  q  not  a  per- 
fect square,  it  is  the  object  of  the  second  transformation  to  len- 
der the  denominator  a   rational  quantity. 


CHAP,  v.]  RADICALS    OF    THE    SECOND    DEGREE.  129 

Tills  object  is  attained  by  multiplying  both  terms  of  the  frac- 
tion by  p  —  y  q ,  when  the  denominator  is  p  +  y  y  ,  and  by 
p  +  \  q ,  Avhen  the  denominator  is  p  —  y  q  '■,  and  recollecting 
that  the  sum  of  two  quantities,  multiplied  by  their  difference,  is 
equal  to  the  difference  of  their  squares :     hence, 

g ^(?  —  Vq)        __  (i{p  —  y/~q)  _  gp  —  a  \/T 

p  +  ^^q      (p  +  VT)  (p  —  Vq)        p'^  -q  p^  —  q' 

a  _  a  (p  -f-  Vq)  _  g(p  +  Vq)  __  ap  +  a  y/q 

p  —  V  q      {p-  Vq)  (p  +  VT)        p^-q  p'^  —  q' 

in  which  the  denominators  are   rational. 

As  an  example  to  illustrate  the  utility  of  this  method  of  ap- 
proximation, let  it  be   required   to    find  the   approximate  value   of 

7 

the  expression    -=rr.     We  write 

3  —  V  5 

7  7(3  +  y/~E)       21  +  7  V^ 


3  —  V5  9  —  5  4 

But,  7  -y/  5   =  V'49  X  5  =  ^245  =  15+    an  irrational 

number  less  than  unity.     Therefore, 

7  21  +  15-1-   irr.  number   <  1  1 


3  —  V  5  4  4 

hence,   9   differs   from  the  true   value   by  less   than  one  Jourth. 

If  we  wish  a  more  exact  value  for  this  expression,  extract  thi 
square  root  of  245  to  a  certain  number  of  decimal  places,  add  21 
to   this   root,  and  divide  the  result   by  4. 

For  another  example,  take 

7  v^5 


and  find  its  value  to  within  0.01. 
We  have, 

7  ^/T    _  7  -/T(  \/n  —  ^/T)  _  7  y/sB  —  7  yTs 

V^il  -\-  ^/3  11  —  3  "  8 

9 


130  ELEME^fTS  OF  ALGKBRA.  [CHAP.  V. 

Now,       7  ^/bb  =  -/55  x"49'=  ^2695  =n  51.91,  williin  0.01, 

and  7-/l5  = -y/lS  X  49  =  ^735     ==27.11      -     -     -     ; 

-     therefore^, 

1  ■/b'  51.91  —27.11  _  24.80  _^  ,^ 

/- — == -=  —  —  —  J. 10. 

1 1  +  V  3  8  8 

Hence,  we  have  3.10  for  the  required  result.     This  is  exact  to 

within     . 

800 

By  a  siDiilar  process,  it  will  be  found  that 

q     I     9      /  7 

^_  ""  y  =  2,123,    exact  to  within  0.001. 

5  V  12  -6/5 

Remark. — The  value  of  expressions  similar  to  those  above,  may 
be  calculated  by  approximating  to  the  value  of  each  of  the  rad- 
icals which  enter  the  numerator  and  denominator.  But  as  the 
value  of  the  denominator  would  not  be  exact,  we  could  not  deter- 
mine the  degree  of  approximation  which  would  be  obtained,  where- 
as by  the  method  just  indicated,  the  denominator  becomes  rational, 
and  we  always  know  to  what  degree   the  approximation  is  made. 

Examples  in  tJie   Calculus  of  Radicals. 

We  make  the  reductions  in  the  examples  which  follow,  accord 
ing  to  the  methods  indicated  in  Art.  125  ;  though,  it  is  sometimes 
necessary  to  multiply  the  quantity  under  the  radical   sign,  instead 
of  dividing  it. 

1.  Reduce     y  125    to  its  simplest  form. 

We  first  seek  the  largest  perfect  square  that  will  exactly  di- 
vide 125.  We  try,  4,  9,  16,  25,  36,  49,  64,  81,  100,  121,  and 
144.  AVe  find  that  25  is  the  only  one  that  will  give  an  exact 
quotient :    hence, 

'/l25  =  -v/25  X  5  =  5  ^/l>  . 

2.  Ke-Juce     \/ to  its  simplest  form. 

We   observe   that  25   will   divide   the   numerator,   and   hencti 


^^    147        V       1 


X  2       ,.  /  2 
—  5 


47  ^    147 


CHAP.   V.l         EXAMPLES    IN    THE    CALCULUS    OF    RADICALS.  iol 

Since  there  is  no  perfect  square  which  will  divide  147,  we 
must  see  if  we  can  multiply  it  by  any  number  which  will  give 
a  perfect  square  for  a  product.  Multiplying  by  2  we  have  284, 
which  is  not  a  perfect  square.  Then  trying  3,  we  find  the  prod- 
not  441,  whose  square  root  is  21.     Hence,  we  have 

Ai"  /  2  X  3  r\  .5      r- 

5  V  - — -  =  5  V =  5\/ xG=  —  Ve 

^   147  V   147  X  3  V  441  ^  2\J^ 

3.  Reduce     -y/oSa^a;    to  its  most  simple  form. 

Ans.    la  yJ'Zx 


4.  Reduce     y  (a;^  —  a^x^)    to  its  most  simple  form. 

5.  Required  the  sum  of    -y/72     and    ^J  128  . 

Ans.    14  V^ 

6.  Required  the  sum  of    y  27    and    y  147  . 

Ans.  10  yy. 


2  /27 

7    Required  the  sum  of   V/  — -    and    \    — . 
4  V    o  V  5Q 


19     r-^ 

Ans.    -— -  y  b 


8.  Required  the  sum  of    2  yj a^h     and    3  ^J^^hx^. 

9.  Required  the  sum  of    9  ^243    and     10  "/3G3. 

/~3~       r^ 

10.  Required  the  difference  of  \    —    and    \    — . 

^  V    5  V  27 

4      r- 
Ans.    — yl5. 
45^ 

11.  Required  the  product  of    5y  8     and    3^5. 

Anjs.    30^10. 

2       / 1  3       /~7 

12.  Required  the  product  of    ~t-\I  -—    and     — \    — . 

^  ^  3^8  4    V  10 

Ans.   l/i^ 

13.  Divide    6 -/lO    by    3-/ 5. 

14.  What  is  the  sum  of    ^J \%alr   +  h  yjlba. 

15.  What  is  the  sum  of    s/ \%a-P  -f  ^50^. 

Ans.    {Sa^-b  +  Sai)  ^/z^ih 


133  ELEMENTS    OF    ALGEBRA.  [CHAP.  VT. 


CHAPTER  VI. 

EQUATIONS    OF    THE    SECOND    DEGREE. 

137.  An  Equation  of  the  second  degree  is  one  in  wliich  the 
greatest  exponent  of  the  unknown  quantity  is  equal  to  2.  If  the 
equation  contains  two  unknown  quantities,  it  is  of  the  second  de- 
gree when  the  greatest  sum  of  the  exponents  with  which  the 
unknown  quantities  are  affected,  in  any  term,  is   equal  to  2. 

138.  Equations  of  the  second  degree  are  divided  into  two 
classes. 

1st.  Equations  which  involve  only  the  square  of  the  unknown 
quantity  and  known  terms.     These  are  called  incomplete  equations. 

2d.  Equations  which  involve  the  first  and  second  powers  of  the 
unknown  quantity  and  known  terms.  These  are  called  complete 
equations. 

Thus,  a;2  +  2x2  —  5  =  7 

and  5a?2  —  Sx^  —  4  =  a, 

are  incomplete  equations  ;    and 

8x2  —  5-p   —  3jj2  -i^  a  ■=:  b 

2x2  —  8x2  —    a;    —  c  =  J, 
are  complete  equations. 

Of  Incomi)lcte  Equations. 

139.  The  following  is  the  most  general  form  of  an  incomplete 
equation  :  viz., 

°^  c  Q  ' 

U  we  reduce  this  to  an  equation  containing  only  entire  terms, 
we  have, 

6ccx2  —  Ghx^  —  7c  rr  Q>cd  • 


CHAP,  VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  133 

hence,  x^  (6ac  —  6i)  =  Qcd  -\-  7c, 

^       Qcd  +  7c 
and,  X-  = —  =  m, 

bac  —  DO 

by  substituting  m,  for  the  known  terms  which  compose  the  sec- 
ond member.  Hence,  every  incomplete  equation  can  be  reduced  to 
an   equation   involving   but   two   terms,   of  the    form 

and  from  this  circumstance,  the  incomplete  equations  are  often 
called,  equations  involving  two  terms. 

There  is  no  difficulty  in  resolving  equations  of  this   form  ;   for, 

we  have  a;  =  y  wi . 

If  m  is  a  perfect  square,  the  exact  value  of  x  can  be  found 
by  extracting  its  square  root,  and  the  value  will  be  expressed 
either   algebraically  or  in   numbers. 

If  m  is  an  algebraic  quantity,  and  not  a  perfect  square,  it  must 
be  reduced  to  its  simplest  form  by  the  rules  for  reducing  radi- 
cals of  the  second  degree.  If  wi  is  a  number,  and  not  a  perfect 
square,  its  square  root  must  be  determined  approximatively  by  the 
rules  already  given. 

But  the  square  of  any  number  is  +,  whether  the  number  it- 
self have  the   -j-   or  —  sign  :    hence,  it  follows  that 

(+  v  OT  )^  =  rn,    and     {—  y  m)'^  :=  m  ; 

and  therefore,  the  uidtnown  quantity  x  is  susceptible  of  two  dis- 
tinct values,  viz. 

X  =z  -\-  y  m  ,    and    x  =  —  y  m  ; 

and  either  of  these  values  being  substituted  for  x  will  verify  the 
piven  equation.     For, 

xXx  =  x'^  =  -\-yrn   X+  ym  =  m;    ' 
and  a:Xa:  =  a;2=  —  ym    X  —  y  m  z::i  m. 

The  root  of  an  equation  is  any  expression  which,  being  substi- 
tuted for  the  unknown  quantity,  will  satisfy  the  equation  ;  that  is, 
make  the  two  members  equal  to  each  other.  Hence,  every  incom- 
plete equation  of  the  second  degree  has  two  roots  which  are  numen- 
cally  equal  to  each  other;  one  having  the  sign  plus,  and  the  other 
the  sisn  minus. 


134  ELEMENTS    OF    ALGEBRA.  [CHAP.  VI. 

1.  Let  US  take,  as  an  example,  the  equation 

3  ^  12  24  ^  24 

which,  by  making  the  terms  entire,  becomes 

8a:2  —  12  +  IQx"^  —  1  —  24a'2  +  299, 
and  by  transposing  and  reducing 

42a;2  =  378      and      x"^  = =  9  ; 

42 

hence,       a;=4-v9   =  +  3;    and    ir^— y9   =—3. 

2.  As  a   second   example,   let  us  take  the  equation 

3a2  =  5. 
Dividing  by   3   and  extracting  the   square  root,  we  have 

T  1 


15; 

3  3    '        ' 

in  which  the  values  must  be   determined  approxiraalively. 

3.  What  are  the  values  of  x  in  the  equation 

11  (a:2  — 4)  =  5(a;2  +  2).  Ans.  x  =  ±Z. 

4.  What  are  the  values  of  x  in  the  equation 

y /7i2  —  a;2  m 
=  n          Ans.  X  ■=■  ±      .  — 

X  Y  1  4-  »^ 

Of  Comj)lete  Equations  of  the   Second  Degree. 

140.  The  most  general  complete  equation  of  the  second  degree 
can  be  expressed  under  the  form 

oa;2  -\-  hx  —  3a;  —  c  =  d ; 
which  may  be  put  under  the  form 

ax"^  +  {h  —  3)x=  d-\-c\ 
and  by   dividing  by  the  co-efficient  of  ar^,  we  have 
^       J  -3  d+  c 

X^  -j X  =  . 

a  a 

If  now,  we  substitute  2p  for  the  co-efficient  of  x,  and  represciii 
the  value  of  the  second  member  by  q,  we  shall  have, 

a;2  -\-  2px  =  q. 


CHAP.   VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  135 

The  reduction  to  the  above  form   is   made  : 

1st.  By  transposing  all  the  terms  involving  x^  and  a;  to  the 
first  member,  and  all  the  known  terms  to   the   second. 

2d.  If  the  term  involving  x^  should  be  negative,  the  signs  of 
all  the  terms  of  the  equation  must  be  changed  lo  render  it  posi- 
tive, and  then  divide  both  members  by  the  co-efficient  of  x^.  Hence, 
every  complete  equation  of  the  second  degree  can  be  reduced  to  aji 
equation  involving  but  three  terms,  and  of  the  above  form.  The 
quantity  q  is  called  the  absolute  term. 

If  we   compare   the  first  member  of  the  equation 

cc?  -\-  2px  =  q, 
with  the  square   of  a  binomial 

{x  -J-  a)-  =  x"^  +  Sax  +  a^ 

we  see  that  it  needs  but  the  square  of  p  to  render  it  a  perfect 
square.  If  then,  p^  be  added  to  the  first  member,  it  will  become 
a  perfect  square  ;  but  in  order  to  preserve  the  equality  of  the 
members,  p^  must  also  be  added  to  the  second  member.  Making 
these  additions,  we  have 

x^  H-  2px  +  p2  =  y  -|-  p2  J 

this  is  called  completing  the  square,  which  is  done  by  adding  the 
square  of  half  the  co-efficient  of  x  to  both  members  of  the  equa- 
tion. 

Now,   if  we  extract  the  square  root  of  both  members,  we  have, 


X  -\-  p  =  ±  y/q  + 
and   by   transposing  p,   we   shall  have 


i2 


a;  =  —  j9  +  ^/q  +  p-,    and    a;  =  —  p  —  yj q  -f  p^. 
Either  of  these  values  being  substituted  for  x  in  the  equation 

x^  +  2^^  =  ? 
will   satisfy  it.     For,   we  have  from  the   first  A'alue, 

ar2  =  (—  p  +  ^JY+fY  _    pi  _2py/q  ^p-i  -\.q  ^  ^,2 

and 
•Ipx  =2p  X  i-p+  yjq  +  p2)    ==  _  2/  +  2p  y/ q  +  f 

hence  x"^  +  2pa:  ~  q. 


136  ELEMENTS    OF    ALGEBKA.  [CHAP.  VJ. 

For  the  second   value,  we  have 


a-2  =  ( —  p  —  ^q  -Y  f-f  —         p^  +2p  y/q  +  p'^  +  q  +  p^ 
and 


2px  =  2p{-p-y/q  +  /)    ^  _  2;,2  _  2p  y/q  +  p2  ; 

hence,  a;^  +  2pa;  =:  q  ; 

and  consequently,   the  values   found  above,  are  roots  of  the   equa- 
tion. 

In  order  to  refer  readily,  to  either  of  these  values,  we  shall  call 
ihe  one  which  arises  from  using  the  +  sign  before  the  radical, 
llie  frst  value  of  .r,  or  the  first  root  of  the  equation  ;  and  the 
other,  the  second  value   of  r,   or   the   second  root  of  the   equation. 

Having  reduced  a  complete  equation  of  the  second  degree  tx) 
the  form 

af2  -\-  2px  =:  q, 

we   can   write  immediately  the  two   values  of  the   unknown  quan- 
tity by  the   following 

RULE. 

The  first  value  of  the  unknown  quantity  is  equal  to  half  the  co- 
efficient of  X  taken  with  a  contrary  sign,  plus  the  square  root  of 
the  absolute  term  increased  by  the  square  of  half  the  co-e£icient  of  x. 

The  second  value  of  the  unJinoiun  quantity  is  equal  to  half  the 
to-ejident  of  x  taken  with  a  contrary  sign,  minus  the  square  root 
vf  the  absolute  term  increased  by  the  sqtiare  of  half  the  co-efficient 
of  X. 

1.  Let   us   take   as   an   example   the  equation 
a:2  —  7a:  +  10  =  0. 

By  transposing    10,    we   have 

a:2  —73;  =:   _  10. 

Hence,         a;  =  3.5  +  y/ —  10  +  (3.5)2  _  3,5  _|_  ^  2.25  =  5, 


and  a;  =  3.5  -  y/ —  10  -f  (3.5)2  ^  3,5  _  ^2.25  =  2. 

2.  As   a   second   example,   let  us   take   the   equation 

5,1  3  2  ,273 

--a:2 X  A =  8  —  —  x  —  a:2-j . 

6  2  4  3  12 

K educing  to  entire  terms,  we  have 

lOa'2  —  6x  4-  9  =  96  -  Sx  -  12:^2  ^  273, 


CHAP.  VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  137 

and  transposing  and  reducing, 

22x2  +  2a;  =  360, 
and  dividing  both  members  by  22, 

„    ,    2  3G0 

22  22  ' 


hence,  -=-^+\/^  +  Q' 

and  .=  -l-\/-  +  (-y 

22        V    22         \22/ 


It  often  occurs,  in  the  solution  of  equations,  that  p^  and  q  are 
fractions,  as  in  the  above  example.  These  fractions  most  gen- 
erally arise  from  dividing  by  the  co-efRcient  of  x'^  in  the  reduc- 
tion of  the  equation  to  the  required  form.  When  this  is  the  case, 
we  readily  discover  the  quantity  by  which  it  is  necessary  to  mul- 
tiply the  terms  of  q,  in  order  to  reduce  it  to  the  same  denominator 
w\ith  p"^ ;  after  which,  the  numerators  may  be  added  together  and 
placed  over  the  common  denominator.  After  this  operation,  the 
denominator  will  be  a  perfect  square,  and  may  be  brought  from 
under  the  radical  sign,  and  will  become  a  divisor  of  the  square 
root  of  the  numerator. 

To  apply  these  principles  in  reducing  the  radical  part  of  the 
values  of  x,  in  the  last  example,  we  have 


/360        /ly  _      /360  X  22  1      _      /792 

V    22   "*"  \22/    ~    V       (22P      +  C22)2  "~  V  ~T 


0+  1 


(22)2        '    (22)2        V       (22)2 

22  ^  22 ' 

and  therefore,  the  two  values  of  x  become, 
_        1        89  _  88  _ 
^  ~  ~22       22  ~  22  ~  "*  ' 

d  —  _J__^__^^_       "^^ . 

*~~~22~'2""~~'22~~lT' 

either  of  which  values  being  substituted  for  x  in  the  given  equa- 
tiim,  will  satisfy  it. 

3.  What  are  the  values   of  x  in  the   equation 

cx2 —  ac  =1  ex  —  bx"^. 


133  ELEMENTS    OF    ALGEBRA.  [CHAP.  VI. 

We   have,  by   transposing  and  reducing, 
[a  -{-  b)  x"  —  ex  =  ac; 
„  c  (IC 


hence,  x^  . 

a  -\-  0  a  -\-  0 

and  consequently, 


/    ac 


2{a-\-b)^^a  +  b'4(a  +  bf 


c  I     (IC  c^ 

If  now,  we  multiply  both    terms  of  y  by  4  (a  +  b),  it  will    be 
reduced  to   a   common   denominator  with  p^,  and   we  shall  have 


v/ 


ac  c^  Aa^c  -\-  \ahc-\-c'^         y/ Ad^c -\- Aabc -{- c^ 

oTT  "^  4  (a  +  bf  ^  ^       4  (a  +  ^)^        ^  2  (a  +  ^')         ' 

c  ±  -sf  Acre  +  4aic  +  c^ 

hence,  a-  = ^-- — ; — -r . 

2  (a  +  6) 

1-.  Wha^  Lre  the  valuer  tf  a",  in  the  'iqaations, 
6x3  _  37^  _  _  57. 

By  reducing  to  the  required   form,   we   have 
2_37  ^  _57_ 
^         G  ^  ~        0  ' 


37  /      57        /37\2 


hence,  x  ■=  ■{ ±\/ — +  ( — ) 

We    observe,    that  if  we    multiply  both   terms    of  q   by  2,   and 
then  by  12,  that  q  and  -p^  will  have  the  same  denominator ;  hence. 


12        V         (12)2        ^  (12)2 
But,  114x12^=1368;    and    (37)2=1369 


.  37    /_  1368  +  1369     37   1 
hence,     .=  +-±V (l^j^— =  +  I^  ^  Tl^  ' 

.  37  ,  1    38   19 

ipnce,     ar=  -| =:  —  z=  — , 

12   12   12    6  ' 

.37    1    36   ^ 

and       X  z=  A =  —  ==3. 

12   12   12 


CHAP.  VI  J  EQUATIONS    OF    THE    SKCOND    DEGREE.  139 

5.  What   are   the   values  of  x   in  the   equation 

4a2  _  2x2  ^  2ax  =  18ab  —  I8b\ 

In  this  equation,  the  term  which  contains  the  second  power  of 
the  unknown  quantity  is  negative  ;  and  since  that  term  already 
stands  in  the  first  member  of  the  equation,  it  can  only  oe  ren- 
dered positive  by  changing  the  sign  of  every  term  of  the  equa- 
tion.    Doing  this,  transposing,   and   dividing  by   2.  we    have 

x"^  —  ax  =  2a'  —  9ab  +  9b^  ; 


whence,       x  =—  ±\/2a^  —  9ab  +9*2+-^ 
2         ^  4 


9^2; 


and  the  root  of  the  radical  part  is  equal   to   -; 3b.     Hence, 

a        ,3a       „,,       ,  Cx=       2a  — 3b. 

X  =  —  d:  ( 2b) ;     hence,      < 

2        ^2  '  '      lx= 


a  +  3^». 


EXAMPLES. 


X                                    4 
1.  Given,    — 4  —  a:2  —  2a: p-  a;2  =  45  —  3x2,    ^^  fi^j  ^ 

(x=        7.12") 

01 


(  X  =       7.12  ) 
Ans.     <  .  /    to  within  O.C 

(  X  =  —  5.73  3 


2.  Given,    a:2  —  8x  +  10  =  19,    to  find  x. 

Ans.  X  =  9,    and   a?  =  —  1. 

3.  Given,    x2  —  x  —  40  =  170,    to  find  x. 

Arts.  X  =  15,   and   x  =  —  14. 

4.  Given,    3x2  +  2x  —  9  =  yg^    to  find  x. 

Ans.  X  =  5,    and   x  =  —  5|. 

5.  Given,    1x2  -  ix  +  7f  =  8,    to  find  x. 

5 
Ans.  X  =z  Ik,  and  x  = 

6.  Given,    77ix2  —  2mx  y  n  =  nx^  —  mn,    to  find  x. 

Y  mn 


V    .....  V    fin 

Ans.   X 


y  m  —  y  n  y  m  -{-  y   n 


140  ELEMENTS    OF    ALGEBRA.  [CHAP.   VI. 

«    ^-  90  90  27 

7.  Given -— i=  0,    to  find  x. 

X        a;  +  1       a;  +  2 

A  A  5 

Ans.  a;  =  4,   x  = — . 

cic 

8.  Given    ax^ r  =  ex  —  hx^,    to  find  x. 

a  +  0 

.  c  ±:  V  c2  +  4ac 

Ans.   X  = 


2{a-\-b) 

«    ^.  ,   ,        3a2  6a2  +  a&  —  2Z-2        Pa;  ,    , 

9  Given     abx^  H a;  = ,    to  find  x. 

c  c^  c 

2a  —  b  3a  -f  2b 

Ans.  X  = ,   a-  = . 

ac  be 

Tn^x 

10  Given    a^  -\- b"^  —  2bx  -\- x^  =  ——,    to  find  a:.  y 

7r 

n  / 

Ans.  X  =  -r r  (bn  dz  -J  cP-rr^  +  b'^m^  —  a^n^). 

n* —  nv- 

QUESTIONS. 

1.  To  find  a  number  such,  that  twice  its  square  added  to  three 
times   the   number,   shall   be   equal   to   65. 
Let  X  represent  the  number.     Then, 
2x2  4-  3a;  —  65, 


3        23 

3        23 

13 

4          4 

~~  2" 

3  ^  /65    ,     9 

whence,        x= r  —  V":r  +  T7'  = 

4  ^2         lb 

there  fijre, 

3  23 

x=3 1 =5,     and    x  =z  — 

4  4 

Both  these  values  will  satisfy  the  question,  understood  in  its 
algebraic  sense.     We  have, 

2  X  (5)2  +  3x5  =  2x25  +  15  =  05; 

/      13\2  13       169       39       130       ^^ 

and        2(-_)+3x--=---  =  -=C5. 

Suppose  we  had  stated  the  question  thus  : — To  find  a  number 
such,  that  three  times  the  number  subtracted  from  twice  its  square, 
Khali  give  a  remainder  equal  to  65. 


CHAP.   VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  141 

If  we  denote  the  number  by  x    we  shall  have 
2a;2  —  3x  =  65  ; 


23 

4 


3        ^  /65   ,     9         3 
whence,  x  =  -±\/-  +  -  =  - 

^       ^  3        23       13  ,  3        23 

iheretore,      a;  = =  — ,    and    x  = =  —  5, 

4         4         2  4         4 

values   which  differ  from   those  found  before,  only  in   their  signs. 
If  the  last  enunciation  be  understood  in  its  algebraic  sense,  the 

13 

—  5    equally    with   the    -\ will    satisfy    both    the    enunciation 

and  the  equation.  It  is  true  that  the  second  term  —  3a;  will 
be  added  to  the  first  term  ;  for,  the  subtraction  of  3  times  —  5, 
will  give    +15. 

2.  A  person  purchased  a  number  of  yards  of  cloth  for  240 
cents.  If  he  had  received  3  yards  less,  for  the  same  sum,  it 
would  have  cost  him  4  cents  more  per  yard.  How  many  yards 
did  he  purchase  ? 

Let  X  =    the   number  of  yards  purchased. 

240 
Then,  the   price   per  yard  will  be    expressed  by    . 

If,  for  240  cents,  he  had  received  3  yards  less,  that  is,  x  —  3 
yards,  the  price  per  yard,  under  this  hypothesis,  would  have  been 

.p.se„.e.  ..   2^,     B.,  .,  ..e  e^nciaUon  ..s  ..  eo. 

would  exceed  the  first,  by  4  cents.  Therefore,  we  have  the  equa- 
tion 

240         240 

5 =4; 

X  —  6  X 

whence,  by  reducing,         a:^  —  3a;  =  180, 


3        ^  /9     .    ,„^       3  ±  27 

and  a:  =  -±\/x  +  lSO  =  — 2— ' 

therefore,  a;  =  15,    and    xz=  —  12. 

The  value    a:  =  15    satisfies   the   enunciation   understood    in   its 

240 
arithmetical   sense  ;    for,    15   yards    for  240   cents,   gives ,    or 

1 6   cents   for  the   price   of  one  yard,   and   12  yards  for  240  cents, 
gives  20  cents  for  the  price  of  one  yard,  which  exceeds   16  by  4 


142  "  ELEMENTS    OF    ALGEBRA.  [CHAP.   VI. 

The  —  12  will  satisfy  the  question  in  its  aloebraic  sense,  and 
considered  without  reference  to  its  sign,  will  be  the  answer  to 
the  following  arithmetical  question  : — A  person  purchased  a  number 
of  yards  of  cloth  for  240  cents :  if  he  had  paid  the  same  sum  for  3 
i/ards  more,  it  would  have  cost  him  4  cents  less  per  yard.  How 
many  yards  did  he  purchase  ? 

Remark. — In  the  solution  of  a  problem,  both  roots  of  the  equa- 
tion will  satisfy  the  enunciation,  understood  in  its  algebraic  sense. 

If  the  enunciation,  considered  arithmetically,  admits  of  a  double 
interpretation,  when  translated  into  the  language  of  Algebra,  the 
solution  of  the  equation  will  make  known  the  fact :  and  hence, 
while  one  root  resolves  the  question  in  its  arithmetical  sense,  the 
other  resolves  another  similar  question  also  in  its  arithmetical 
sense  ;  and  both  questions  will  be  stated  by  equations  of  the  same 
general  form,  having  equal  numerical  roots  with  contrary  signs. 

3.  A  man  bought  a  horse,  which  he  sold  for  24  dollars.  At 
the  sale,  he  lost  as  much  per  cent,  on  the  price  of  his  purchase, 
as  the  horse  cost  him.     What  did  he  pay  for  the  horse  ? 

Let  X  denote  the  number  of  dollars  that  he  paid  for  the  horse : 
then,  X  —  24  Avill  express  the  loss  he  sustained.     But  as  he  lost  x 

X 

per  cent,  by  the  sale,  he  must  have  lost  -—  upon  each  dollar,  and 

upon  X  dollars  he  loses   a  sum  denoted  by    — — ;    we  have   then 
the  equation 


a;2 

=  a;  —  24,    whence    a;2  —  100a:  =  —  2400  ; 

100  ' 

and  a-  =  50  ±  -/—  2400  +  2500  =  50  rb  10. 

Therefore,  a;  =  GO    and    x  =  40. 

Both  of  these  values  satisfy  the  question. 

For,  in  the  first  place,  suppose  the  man  gave  60  dollars  for 
the  horse  and  sold  him  for  24,  he  then  loses  36  dollars.  Bui, 
from  the  enunciation  he  should    lose  60  per  cent,  of  60,  that  is, 

of  60    = =     36  ;    therefore    60   satisfies  the  enun- 

100  100 

elation. 

If  lie   pays   40   dollars   for  the  horse,  he  loses   16  by  the  sale; 


CHAP.    VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  143 

40 

for,  lie  should   lose  40  percent,  of  40,  or    40  x  =  16;   thcre- 

^  100 

fore  40  verifies  the  enunciation. 

4.  A  grazier  bought  as  many  sheep  as  cost  him  jC60,  and  after 
reserving  15  out  of  the  number,  he  sold  the  remainder  for  £54, 
and  gained  2s.  a  head  on  those  he   sold  :  how  many  did  he  buy  ' 

Ans.  75. 

5.  A  merchant  bought  cloth  for  which  he  paid  jC33  15,y.,  which 
he  sold  again  at  £2  S.y.  per  piece,  and  gained  by  the  bargain 
as  much  as  one  piece  cost  him :    how  many  pieces  did  he  buy  ? 

Ans.  15. 

6.  What  number  is  that,  which,  being  divided  by  the  product 
of  its  digits,  the  quotient  is  3  ;  and  if  18  be  added  to  it,  the 
digits  will  be  inverted  ?  Ans.  24. 

7.  To  find  a  number  such  that  if  you  subtract  it  from  10,  and 
multiply  the  remainder  by  the  number  itself,  the  product  shall  be  21. 

Ans.  7  or  3.  - 
S.  Two  persons,  A  and  B,  departed  from  different  places  at  the 
same  time,  and  travelled  toward  each  other.  On  meeting,  it  ap- 
peared that  A  had  travelled  18  miles  more  than  B  ;  and  that  A 
could  have  gone  B's  journey  in  15|  days,  but  B  would  have  been 
28  days   in  performing   A's  journey.     How  far  did   each  travel  ? 

A  72  miles. 
B  54  miles. 

9.  A  company  at  a  tavern  had  X8  lbs.  to  pay  for  their  reck- 
oning ;  but  before  the  bill  was  settled,  two  of  them  left  the  room, 
and  then  those  who  remained  had  10.s.  apiece  more  to  pay  than 
before:  how  many  were  there  in  the  company?  Aiis.  7. 

10.  What  two  numbers  are   those   whose    difference   is   15,  and*" 
of  which  the  cube  of  the  lesser  is  equal  to  half  their  product  ? 

Ans.  3  and  18. 

11.  Two  partners,  A  and  B,  gained  $140  in  trade:  A's  money 
was  3  months  in  trade,  and  his  gain  was  S60  less  than  his  stock  ; 
B's  money  was  S50  more  than  A's,  and  was  in  trade  5  months  • 
what  was  A's  stock?  Ans.  $100. 

12.  Two  persons,  A  and  B,  start  from  two  different  points  and 
travel  toward  each  other.  When  they  meet,  it  appears  that  A  has 
travelled  30  miles  more  than  B.     It  also  appears  that  it  M'ill  take  A 


Ans. 


144  ELEMENTS    OF    ALGEBRA.  [CHAP.  VI 

4  days  to  travel  the  road  that  B  had  come,  and  B  9  days  to  travel 
the  road  that  A  had  come.  What  was  tlieir  distance  apart  when 
they  set  out?  Ans.   150  miles. 

Discussion  of  Equations  of  the   Second  Degree. 

141.  Thus  far,  we  have  only  resolved  particular  problems  in- 
volving equations  of  the  second  degree,  and  in  which  the  known 
quantities  were  expressed  by  particular  numbers. 

We  propose  now,  to  explain  the  general  properties  of  these 
equations,  and  to  examine  the  residts  which  flow  from  all  the  sup- 
positions that  may  be  made  on  the  values  and  signs  of  the  known 
quantities  which  enter  into  them. 

142.  It  has  been  shown  (Art.  140),  that  every  complete  equa- 
tion of  the  second  degree  can  be  reduced  to  the  form 

x^  +  2px  =  q  (1), 

in  which  p  and  q  are  numerical  or  algebraic  quantities,  whole 
numbers,  or  fractions,   and  their   signs   plus   or  minus. 

If  we  make  the  first  member  a  perfect  square,  by  adding  p"^  to 
both  members,  we  have 

x^  +  2j9a;  -\-  p"^  =.  q  -{-  p^, 
which  may  be  put  under  the  form 

{x  -^  pY  =  q  -\-  p^. 

Whatever  may  be  the  value  of  ^  -}-  V^i  ^^^  square  root  may  be 
represented  by  m,  and  the  equation  put  under  the  form 

{x  +  py-  =  771^,    and  consequently,    {x  +  pY  —  m'  ==  0. 

But,  as  the  first  member  of  the  last  equation  is  the  difference 
between  two  squares,  it  may  be  put  under  the  form 

(x  +  /)  —  n?)  (a;  +  p  +  to)  =  0,  (2) 

in  which  the  first  member  is  the  product  of  two  factors,  and  the 
second  0.  Now  we  can  make  the  product  equal  to  0,  and  con- 
sequently satisfy  equation  (2),  only  in  two  diflerent  ways :  viz., 
making 

X  -\-  p  —  /«  =  0,    whence,    a:  =  —  p  +  »i, 
.ir,  by  making 

a;  +  p  -f  "^  =  0,    w'honce,    x  ■=.  --  p  —  m  ; 


CHAP.  VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  145 

and  by  substituting  for  m  its  value,  we  have 

X  =  —  ;?  +  \q  +  Jp--,    and    r  =  —  p  —  yy  +  V^- 

Now,  either  of  tliese  values  being  substituted  for  x  in  its  cor- 
responding factor  of  equation  (2),  will  satisfy  that  equation,  and 
consequently,  will  satisfy  equation  (1),  from  which  it  was  derived 
Hence  we  conclude, 

1st.  That  every  equation  of  the  second  degree  has  two  roots,  and 
only  two. 

2d.  That  every  equation  of  the  second  degree  may  he  decomposed 
into  two  binomial  factors  of  the  first  degree  with  respect  to  x,  having 
X  for  a  frst  term,  and  the  two  roots,  taken  with  their  signs  changed, 
for  the  second  terms. 

For  example,  the  equation 

a;2  +  3a;  —  28  =  0 

being  resolved  gives  a:  =  4   and  a;  =  —  7  ;  either  of  wliich  values 
will  satisfy  the  equation.     We  also  have 

(x  -  4)  (x  +  7)  ==  x2  +  3a;  —  28. 

If  the  roots  of  an  equation  are  known,  we  readily  form  the  bi- 
nomial factors  and  the  equation. 

1 .  What  are  the  factors,  and  what  is  the  equation,  of  which 
the  roots  are  8  and  —  9  ? 

X  —  8    and    a;  +  9 
are    the  binomial  factors,  and 

(a;  —  8)  (  a-  +  9)  =  ;t2  -f  a?  —  72  =  0 
is  the   equation. 

2.  What  are  the  factors,  and  what  is  the  equation,  of  which 
the  roots  arc;   —  1    and  +  1. 

[x  -{-  l)(r  —  1)  =:a;2—  1  =0. 

3.  What  are  the  factors  and  what  is  the  equation,  whose  roots 
are 

7  +  J  —  1039         ,     7  —  J  —  1039 

— —    and    — — . 

16  16 

—  8x2  _  7a;  4.  34  =  0. 
10 


146  ELEMENTS    OF    ALGEBRA.  [CHAP.    VI. 

1 1,3    If  we  designate   llie  two  roots  of  any  equation  by  x'  ajid 
x" .  we  shall  have 


3?'  =  —  p  +  -yj q  +  j)2^    and  otf'  =.  —  p  ~-  \q  +  p^ ; 
by  adding  the  roots,  we  obtain, 

x'  +  x''  =  —  2p  ; 
•in, I  by  multiplying  them  together, 

x'x"  =  —  q.     Hence, 

1st.  The  algebraic  sum  of  the  two  roots  is  equal  to  the  co-efficient 
of  the  second  term  of  the  equation,  taken  with  a  contrary  sign 

2d.  The  product  of  the  two  roots  is  equal  to  the  absolute  term, 
taken  also  with  a  contrary  sign. 

141.  Thus  far,  we  have  regarded  p  and  q  as  algebraic  quanti- 
ties, without  considering  the  essential  sign  of  either,  nor  have  we 
at  all  regarded  their  relative  values 

If  we  first  suppose  p  and  q  to  be  both  essentially  positive,  then 
to  become  negative  in  succession,  and  after  that,  both  to  become 
negative  together,  we  shall  have  all  the  combinations  of  signs 
which  can  arise  ;  and  the  complete  equation  of  the  second  de- 
gree will,  therefore,  always  be  expressed  under  one  of  the  four 
following  forms : — 

a;2  +  2px  =  q  (1), 
'J?  —  2px  =  q  (2), 
^2  ^  2px  =  -  q  (3), 
x"^  —  2px  =  —  q  (4). 

These  equations  being  resolved,  give 

x=  ~  p  ±  y/       q+  p^  (1), 


x~  +pr^^/      q+p-^  (2), 

x=  —p  ±  y/  —q  +  p^  (3), 

a;  =  +  p  rt  V  —  q-\-  p^  (4). 

In  order  that  the  value  of  x,  in  these  equations,  may  be  found, 
either  exactly  or  approximatively,  it  is  necessary  that  the  quan- 
tity under  the  radical  sign  be  positive  (Art.   126). 

Now,  p^  being  necessarily  positive,  whatever  may  be  the  sign 
of  />,  it  follows,   that  in  the  first  and  second   forms   all  the  values 


CHAP.   VJ.]  EQUATIO.VS    OK    THK    SECO.VD    DEGREE.  1  t7 

of  X  will  be  real.  They  will  be  determined  exactly,  when  the 
quantity  q -\- p'^  is  a  perfect  square,  and  approximatively,  when  it 
is  not  so. 

Since  q  and  p"^  are  both  positive,  the  numerical  value  of  the 
radical  expression  ±  -y/q  +  p'^  will  be  greater  than  p,  and  hence 
the  second  member  of  the  equation  will  have  the  same  sign  as 
the  radical.  Therefore,  in  the  first  form,  the  first  root  of  the  equa- 
tion will  be  positive,  and  the  second  root  negative. 

The  positive  root  w^ll,  in  general,  as  already  observed,  alone 
satisfy  the  problem  understood  in  its  arithmetical  sense  ;  the  neg- 
ative value,  answering  to  a  similar  problem,  differing  from  the  first 
only  in  this ;  that  a  certain  quantity  Avhich  is  regarded  as  addi- 
tive in  the  one,  is  subtractive  in  the  other,  and  the  reverse. 

In  the  second  form,  the  first  value  of  x  is  positive,  and  the 
second  negative,  the  positive  value  being  the  greater. 

In  the  third  and  fourth  forms,  the  values  of  x  will  be  imaginary 
when    q  >  p^,    and   real  when    q  <  p"^. 

And  since  y  —  q -\- p'^  <C  p,  it  follows  that  the  real  values  of 
X  will  both  be  negative  in  the  third  form,  and  both  positive  in  the 
fourth. 

145.  The  last  properties  which  have  been  proved,  may  be  shoAvn 
from  the  two  properties  of  an  equation  of  the  second  degree,  de- 
monstrated in  Art.   143.     They  are  : 

The  algebraic  sum  of  the  roots  is  equal  to  the  co-efiicicnt  of  the 
second  term,  taken  with  a  contrary/  sign,  and  their  product  is  equal 
to  the  absolute  term,  taken  also  with  a  contrary  sign. 

In  the  first  two  forms,  q  being  positive  in  the  second  member, 
it  follows  that  the  product  of  the  two  roots  is  negative  :  hence, 
they  will  have  contrary  signs.  But  in  the  third  and  fourth  forms, 
q  being  negative  in  the  second  member,  it  follows  that  the  prod- 
uct of  the  two  roots  wi'll  be  positive  :  hence,  they  will  have  like 
signs,  viz.,  both  negative  in  the  third  form,  where  2p  is  positive, 
and  both  positive  in  the  fourth  form,  where  2p   is   negative. 

Moreover,  since  the  sum  of  tlie  roots  is  affiscted  with  a  sign 
contrary  to  that  of  the  co-efficient  2p,  it  follows  that  the  nega- 
tive root  will  be  the  greatest  in  the  first  form,  and  the  least  in  the 
second. 


2^2 

the  greater  part  (Art.  32). 

a         d  _ 
2         2  ~ 

the  less  part. 

o2           (f2_ 

4         4   ~ 

P, 

their  product  (Art.  46). 

148  ELEMENTS    OF    ALGEBRA.'  [CHAP.  VI. 

146.  We  will  now  show,  that  when  in  the  third  and  fourth 
forms  the  roots  become  imaginary,  that  is,  when  q  >  p^,  that  the 
conditions  of  the  question  will  be  incompatible  with  each  other, 
and  therefore,  the  values  of  x  ought  to  be  imaginary. 

Before  showing  this,  it  will  be  necessary  to  establish  a  propo- 
sition on  which  it  depends :  viz., 

If  a  given  number  be  decomposed  into  two  parts,  and  those  parts 
multiplied  together,  the  product  will  be  the  greatest  possible  when  the 
parts  are  equal. 

Let  a  be  the  number  to  be  decomposed,  and  d  the  difference 
of  the  parts.     Then 


and 


and 

Now  it  is  plain,  that  P  will  increase  as  d  diminishes,  and  that 
it  will  be  the  greatest  possible   when    ^  =  0  ;    that  is, 

a         a         a^     .      ^  . 

—  X  —  =  —    IS  the  greatest  product. 

147.  Now,  since  in  the  equation 

x^  —  2px  =  —  q 

2p  is  the  sum  of  the  roots,  and  q  their  product,  it  follows  that  q 
cannot  be  greater  than  p'^.  The  relations  between  p  and  q,  there- 
fore, fix  a  limit  to  the  value  of  q  ;  and  if  we  assume,  arbitrarily, 
q  >  p",  we  express  by  the  equation  a  condition  which  cannot  be 
fulfilled,  and  this  contradiction  is  made  apparent  by  the  values  of 
X  becoming  imaginary.     Analogy  would  lead   us  to  conclude  that. 

When  the  value  of  the  unknown  quantity  is  found  to  be  imagin- 
ary, the  conditions  expressed  by  the  equation  are  incompatible  tvith 
each  other. 

Remark. — Since  the  roots,  in  the  first  and  second  forms,  have 
contrary  signs,  the  condition  that  their  sum  shall  be  equal  to  a 
given  number  2p,  does  not  fix  a  limit  to  their  product :  hence,  in 
those  two  forms  the  roots  are  never  imaginary. 


CHAP.   VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  149 

148.  We    shall    conclude    this    discussion    by  the   following  re 
marks : — 

1st.  If,  in  the  third  and  fourth  forms,  we  suppose  q  =i  p^,  the 
radical  part  of  the  two  values  of  x  becomes  0,  and  both  the  values 
reduce  to    a;  =   =F  p  :    the  two  roots  are  then  said  to  he  equal. 

In  fact,  by  substituting  p^  for   q  in  the   equation,   it  becomes 
x^  ±  2px  =:  —  p'^,    whence 
a:2  ±  2px  +  p2  —  0,    that  is,     (x  ±  pf  =  0. 

Under  this  supposition,  the  first  member  becomes  the  product 
of  two  equal  factors.  Hence,  the  roots  of  the  equation  are  equal, 
since  the  two  factors  being  placed  equal  to  zero,  give  the  same 
value  for  x. 

2d.  If,  in  the  general  equation, 

X-  +  2px  =  q, 

we  suppose  y  =  0,  the  two  values  of  x  reduce  to 

X  =  —  p  -\-  p  =1  0,    and    x  =:  —  p  —  p  =  —  2p. 
Indeed,  the  equation  is  then  of  the  form 

x2  +  2px  =0,    or    x{x  +  2p)  =  0, 
which  can  only  be  satisfied,  either  by  making 
X  =  0,      or      a?  +  2p  =  0  ; 
whence,  x  =  0,    and    x  =  —  2p ; 

that  is,  one  of  the  roots  is  0,  and  the  other  the  co-efEcient  of  x, 
taken  with  a  contrary  sign. 
3d.  If,  in  the  general  equation 

x"^  ±  2px  z=z  ±  q, 
we  suppose  2p  =  0,  there  will  result 

a;2  =  rfc  y,    whence,    x  =  ±  y  dz  q ; 
that  is,  in  this  case  the  two  values  of  x  are  equal,   and   have   con- 
trary signs,  real   in  the  first   and   second  forms,   and   imaginary  in 
the  third  and  fourth. 

The  equation  then  belongs  to  the  class  of  equations  involving 
two  terms,  treated  of  in  Art.  139. 

4th.  Suppose  we  have  at  the  same  time  j9  1=  0,  5'  =  0 ;  the  equa- 
tion reduces  to  x"^  z=i  0,    and  gives  two  values  of  x,  equal  to  0. 


150  ELEMENTS    OF    ALGEBRA.  [CHAP.   VI, 

149.  There  remains  a  singular  case  to  be  examined,  which  is 
often  met  with  in  the  resolution  of  problems  involving  equations 
of  the  second  degree. 

To  discuss  it,  take  the  equation 

ax^  -\-  bx  =  c, 

,.,       .  —  b  ±  y/b'^  +  4ac 

which  gives  x  =z . 

^  2a 

Suppose,  now,  that  from  a  particular  hypothesis  made  upon  the 

given  quantities  of  the  question,  we  have    0  =  0;    the  expression 

for  X  becomes 

_  0 

b±b       ,  I  ^'^T' 

whence. 


0       '       ~'       ]  2b 

^==-¥' 

0 

Let  us  first  interpret  the  first  root  of    x  —  -—. 

By  multiplying  the   numerator    and   denominator  of  the    second 
member  of  the  equation 

b  +  -/^»2  _|_  4ac 


X  = 


by     —  b  —  ■\/b'^  +  4ac 


2a 
we  obtain 

b"^  —  (£2  _(_  4ac)  —  4ac 

2a  (—b  —  y/b'^+  4ac)        2a  {— b  —  y/U^  +  Aac) 


X 


hence,  x  = "^  r,     by  dividing  by  2a 

—  b  —  -y/Z)^  -|-  4oc 

c 
and  consequently,  x  =z  — ,      by  making  a  =  0. 

Hence  we  see  that   the    apparent  indetermination  arises  from  a 
common  factor  in  the  numerator  and  denominator. 
In  regard  to  the  second  root 

2b 

^=--0"' 

we  see  that  it  is  presented  under  the  form  of  infinity.     By  making 
a  =  0,  in  the  equation 

ax"^  -\~  bx  ^=  c, 

it  reduces  to  an  equation  of  the  first  degree, 

bx  =  c. 


CHAP.  VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  151 

It  is  therefore  impossible  that  it  can  hive  two  roots ;  and  hence, 
such  a  supposition  gives  one  of  the  vahies  of  x  infinite. 

We  have  ah-eady  seen  (Art.  147),  that  imaginary  values  of  the 
unknown  quantity  indicate  the  introduction,  into  the  equation,  of 
contradictory  conditions.  By  considering  the  above  discussion,  and 
that  of  Art.  110,  we  would  conclude,  that  a  result  which  is  in- 
finite, indicates  the  introduction  into  the  equation  of  a  condition  thai 
is  absolutely  impossible. 

If  we  had  at  the  same  time 

a  =  0,    b  —  0,    c  =  0, 

the  proposed  equation  would  be  altogether  indeterminate.  This  is 
the  only  case  of  indetermination  that  the  equation  of  the  second 
degree  presents. 

We  are  now  going  to  apply  the  principles  of  this  general  dis- 
cussion to  a  problem  which  will  give  rise  to  most  of  the  circum- 
stances that  are  commonly  met  with  in  problems  involving  equa- 
tions of  the  second  degree. 

Problem  of  the  Lights. 


C"  A  C      B         C 

150.  Find  upon  the  line  which  joins  two  lights,  A  and  B,  of 
different  intensities,  the  point  which  is  equally  illuminated ;  ad- 
mitting the  following  principle  of  physics,  viz :  The  intensity  of 
the  same  light  at  two  different  distances,  is  in  the  inverse  ratio  of 
the  squares  of  these  distances. 

Let  the  distance  AB,  between  the  two  lights,  be  expressed  bv 
c;  the  intensity  of  the  light  A,  at  the  units  distance,  by  a;  that 
of  the  light  B,  at  the  same  distance,  by  b.  Suppose  C  to  be  the 
equally-illuminated  point,  and  make  AC  =:  x,  whence  BC=zc  —  x. 

By  the  principle  we  have  assumed,  the   intensity  of  tI,   at   the 

unity  of  distance,  being  a,  its    intensity  at   the   distances  2,  3,  4, 

.  a         a        a  -,       ■,■ 

cic,  will  be    —-,     -— ,     — -,  &c.:   hence,  at  the  distance  x  it  will 
4        9       16 

be  .expressed  by    — .     In  like  manner,   the    intensity  of  B  at  tlif 

b 

distance    c  —  x,   is ;    but,   by  the    conduions,  these    two 

(c  —  xy 


152  ELEMENTS    OF    ALGEBRA.  [CHAP.  VI. 


a'  A  C      B        c 

intensities   are    equal   to   each   other,  and   therefore  we  have   the 
equation 


a;2       (c  —  xy ' 
wliich  can  be  put  under  the  form 

(c  —  xy_  b 

x^  a ' 


Hence,  =  — j=^  ;  whence, 


c  -J  a  ,  .  ,      .  c  \/  h 

1st  root  IS,    x=. -   ■ — = -j^=,  which  gives,   c  —  a?  = 


ya+yo  ya-|-yo 


^,            •                    cya              i-i-                            —  c  yf  h 
2d  root  IS,    X  =  — ^= -j=.,  which  gives,  c  —  xz=.     j=^ 

■y/  a  —  yi  yf  a  —  yi 

1st.  Suppose     a'^  b. 
i. '  e  first  value  of  x  is  positive ;   and  since 

y^ <i, 

it  will  be  less  than  c,  and  consequently,  the  required  point  C, 
will  be  situated  between  the  points  A  and  B.  We  see  moreover 
that  the  point  will  be  nearer  B  than  A ;  for,  since  a'^h,  we 
have 

ya+ya     or,     2ya>  (y  a  +  y   i),     whence, 


ya                  1             ,                       ,               C'<J  a  c 

>  —  ;    and  consequently,        . — t=  >  — . 


■y/~a   +  y/~b         2'  '     -v/T  +  y*         2 

Indeed,  this  ought  to  be  the  case,  since  the  intensity  of  A  was 
supposed  greater  than  that  of  B. 

The  corresponding  value  of  c  —  x,  as  may  be  easily  shown,  is 
also  positive,  and  less  than  one  half  of  c ;   that  is, 

c 
< 


-/T+  V~b         2 
The  second  value  of  x  is  also  positive  ;   but  since, 


CHAP.  VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  153 

V  a 


a  —  y   6 

it  will  be  greater  than  c ;  and  consequently,  the  required  point 
will  be  at  some  point  C\  on  the  prolongation  of  AB,  and  at  the 
rijiht  of  the  two  lights. 

We  may,  in  fact,  conceive  that  since  the  two  lights  exert  their 
illuminating  power  in  every  direction,  there  should  be  upon  the 
prolongation  of  AB,  another  point  equally  illuminated ;  but  this 
point  must  be  nearest  that  light  whose  intensity  is  the  least. 

We  can  easily  explain,  why  these  two  values  are  connected 
by  the  same  equation.  If,  instead  of  taking  AC  for  the  unknown 
quantity  x,  we  had  taken  AC\  there  would  have  resulted 
BC  =.  X  —  c  \    and  the  equation 

a  b 


Now,  as  [x  —  c)2  is  identical  with  (c  —  xy,  the  new  equation  is 
identical  with  that  already  established,  which  consequently  should 
have  given  AC^  as  well  as  AC. 

And  since  every  equation  is  but  the  algebraic  enunciation  of  a 
problem,  it  follows  that,  when  the  same  equation  enunciates  several 
problems,  it  ought  by  its  different  roots  to  solve  them  all. 

When  the  line  AC^  is  represented  by  the  unknown  quantity  x 
both  members  of  the  equation 

_     _      -c-y/T 
■yj  a   —  Y   6 

are  negative,    as   they  ought  to  be,  since  a;  >  c. 
By  changing  the  signs  of  both  members,  we  have 

=  BC. 


Y  a   —  Y   6 
2d.  Let    a  <  S. 
This  supposition  gives  a  positive  value  for 

c  Y  a 
y/~a  +  ypb^ 

md  since       ■s/a   +  yfb   >  y/~a   -\-  ^/  a,    that  is,    >  2  ■s/~a 


154 

ELEMENTS    OF    ALGEBRA. 

[CHAP.  VI, 

a'~ 

A         C           B 

/ 

it  follows  that, 

c 

and  consequently, 

'-''>-7r' 

and  therefore,  under  this  hypothesis,  the  point  C,  situated  between 
,4  and  B,  will  be  nearer  to  A  than  B,  as  indeed  it  ought,  since 
the  feebler  light  is  at  A. 

The  second  value  of  x,  that  is, 

c  \/  a  c  y  a 


ya  —  y6  y  b   —  ya 

is  essentially  negative.     How  is  it  to  be  interpreted  ? 

Let  us  suppose  that  we  had  considered  C^^,  at  the  left  of  A, 
as  the  point  of  equal  illumination,  and  that  we  had  represented 
AC'  by  -X. 

Then,  BC'  =  BA  +  AC' ; 

that  is,  BC  =  c  -\-  (—  x)  =  c  —  x; 

and  the  equation  of  the  problem  would  be 

a  b  ^        .        a  h 

that  IS,    — -  = 


and  therefore,  this  equation  ought  to  give  the  point  C  which  lies 
to  the  left  of  A,  as  well  as  the  points  C  and  C  which  lie  to  the 
right. 

It  should  be  observed,  that  we  hav^e  regarded  —  x,  which  rep- 
resents AC'\  as  a  mere  symbol,  without  reference  to  the  essential 
sign  of  X.  Indeed,  the  essential  sign  of  the  unknown  quantify  is 
in  general,  only  made  knoion  in  the  final  result. 

If  it  appears,  in  the  final  result,  that  x  itself  is  negative,  the 
numerical  value  of 

BC"=zc  +  {—x)    becomes    BC''  =  c-irX\ 
that  is,  BC  will  be   equal    to  c  plus   the    numerical  value  of  x 
or  to  c  minus  its  algebraic  value.     Hence, 

\  b   —  ya         -^  b   - 
a  quantity  which  is   essentially  positive. 


CHAP.   VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  155 

3d.   Let    a  ::zz  h. 

Under  this    supposition,  the  vahie  of  ar,  and  that  of   c  —  x,    for 

the  point   C  between  A  and  i?,  both  reduce  to   — ;    that  is,  when 

the  lights  are  of  equal  intensity,  the  point  of  equal  illumination  is 
.'U  the  middle  of  the  line  AB. 

The  value  of  x,  and  that  of  c  —  x,  for  the  points  C  and  C'\ 
vvliich  lie  on  the  prolongation  of  AB,  both  reduce  to 

-\-  c  -J  a  —  c  -J  b        .  ■    ^    ■ 

,    or,  to ,    that  is,  to  innnitv ; 

0  0  ' 

which  indicates,  that  the  conditions  of  the  question  are  absoluteli/ 
impossible.  It  is  evident,  indeed,  that  they  are  so ;  for,  when  the 
intensity  of  the  two  lights  is  equal,  no  part  lying  on  the  prolon- 
gation of  AB  could  be  as  much  illuminated  by  the  distant  as  by 
the  nearer  light :  hence,  the  supposition  of  equal  illumination,  from 
which  the  equation  of  the  problem  is  derived,  is  impossible ;  and 
this  is  shown  in  the  analysis  by  the  corresponding  values  of  the 
unknown  quantity  becoming  infinite. 

4th.  Let    a  =z  b,    and    c  =  0. 

Under  these  suppositions,  the  value  of  x  and  of  c  —  a;,  for  the 
point  of  equal  illumination  between  A  and  B,  both  reduce  to  0, 
as  indeed  they  ought  to  do,  since  the  points  A,  B,  and  C,  are 
then  united  in  one. 

The  value  of  x,  and  of  c  —  x,  for  the  points  C^  and  C,  re- 
duce to  the  indeterminate  form 

0 

Resuming  the  equation  of  the  problem 


(a  —  5)  a;2  -  2 


acx 


rCr 


we  see  that  it  becomes,  under  the  above  suppositions, 

0.a:2  —  0.x  =  0, 

A"hir  h  may  be  satisfied  by  giving  to  x  any  value  whatever :  hence, 
t  is  a  case  of  indetermlnation.  Indeed,  since  the  two  lights  are 
if  the  same  intensity,  and  are  placed  at  the  same  point,  they  ought 
0  illuminate  equalli/  every  point  of  the  straight  line. 


156  ELEMENTS    OF    ALGEBRA.  [CHAP.  VI. 

5th.  Let    c  =  0,     and  a  and  b  be  unequal. 
Under  this   supposition,   both   values   of  x,    and   both  values   of 
c  —  X,  will  reduce  to  0 ;  and  hence,  there  is  but  one  point  of  the 
line  that  will  be  equally  illuminated,  and  that  is  the  point  at  which 
the  two  lights  are  placed. 

In  this  case,  the  equation  of  the  problem  reduces  to 
{a—h)x^  =  0, 
which  gives  two  values, 

a;  =  0,    and    x  =z  0. 

The  preceding  discussion    presents    a    striking    example  of  the 

precision  with  which   the    algebraic    analysis   responds    to   all  the 

relations  which  exist   between   the  quantities   that  enter    into  the 
enunciation  of  a  problem. 

Examples  involving  Radicals  of  the  Second  Degree. 

/ 2a2 

1.  Given,    x  -^  ^/ a'^  -\-  x"^  =z  .,    to  find  x. 

y  m2  _^   ja2 

By  reducing  to  entire  terms,  we  have 
X  yj  d^  +  ^"^  -\-  c?  -{-  y?"  ^=^  2a^, 
by  transposing,  x  y  a^  -\-  x"^  ~  a"^  —  x"^, 

and  by  squaring,  a^x"^  +  a:*  =  a*  —  2a-a;2  -f  ar*, 

hence,  Sa^ac^  =  a*. 


and  consequently,  ^ 


/«2 


2.  Given,    V  "^  +  *^  —  V  "^  -b^  =  b,    to  find  x. 


By  transposing,       \J  —  -{- b"^  =:  y  —  — b^  -\-  b  ; 

and  by  squaring,  —  +  ^^  =  —  —  ^^  +  2(^  y  —  —  i^  _[.  J2. 

hence,          52  =  26 \/-^ -  ^2,    and    6  =  2\/-^-62; 
and  by  squaring,  b"^  =  — 4h^  ; 


CHAP.   VI.]  EQUATIONS    OF    THE    SECOND    DEGREE  157 

and  hence,  x^  = ,    and    x  =  dz 


5^2  b  V  5 


^  a  /a^  —  a;^        x 

3.  Given, 1-  V =  -j-,    to  find  x. 

Ans.  X  =  ±1  -y/'iab  —  b'^ 

4.  Given,    s/^+I +  ^^/If.  =  Ps/IZl, 


=v 

1      "• 

a' 

to  find  X. 

Ans. 

X 

a 

{b:^\f 

X. 

Ans.  X  ■ 

^2ay/  h 

1+6 


ft         ,    *»/ /i^   T 

5.  Given,    — r  =  &,    to  find  x. 

a  +  y  a^  —  x^ 


^     ^ .  v  a;   +  v^  —  «  "^«  ^    ■■ 

6.  Given,    -5-^=:= ^^^ = ,    to  find  x. 

ij  X  —  yf  X  —  a       X  —  a 

a  (1  ±  nf 

Ans.  a;  = —. 

1  ±2» 

7.  Given,    -^^—,== 1-         ^—      =  V  ~'    ^°  ^""^  *' 

7ln5.  a?  =  ±  2  \/ai  —  V^. 

ind  a:. 

a(l  IF  V^2&~-^62) 


„     _,.  a  +  a:  4-  v  2ax  +  a:^        ,  ^    , 

8.  Given, =  Z>,    to  find  a: 

a  -\-  X 


Ans.   X 


■y/2b  -  b^ 


Of  Trhiomial  Equations. 
151.  Every  equation  which  can  be  reduced  to  the  form 

a?'"  +  2^0?"  =  q, 

in  which   m   and  n   are    positive    whole   numbers,  and  2p  and  q, 
known  quantities,  is  called  a  trinomial  equation. 

Hence,  a  trinomial  equation  contains  three  kinds  of  terms  :  viz., 
terms  which  contain  the  unknown  quantity  afiected  with  two  dif- 
ferent exponents,  and  one  or  more  known  terms. 


158  ELEMENTS    OF    ALGEBRA.  [CHAP.   VI. 

If  we  suppose  m  =  2    and    n=  1,    the  equation  becomes 
x^  4-  2px  =  q, 
a  trinomial  equation  of  the  second  degree. 

152.  The  resolution  of  trinomial  equations  of  the  second  degree, 
has  already  been  explained,  and  the  methods  which  were  pursued 
are,  with  some  slight  modifications,  applicable  to  all  trinomial  equa- 
tions in  which  m  =  2n,  that  is,  to  all  equations  of  the  form 

a:-"  +  2px''  =  q. 

Let  us  take,  as  an  example,  the  trinomial  eijuation  of  the  fourth 
degree, 

ex''  —  dx^  —  ax*  +/=  7  +  h. 

We  have,  x'  -\ x'^  = ; 

c  —  a  c  —  a 

and  by  substituting  2p  for  the  co-efTicieiit  of  x"^,  and  q  for  the  ab- 
solute term,  we  have 

x'^  +  2px'^  =z  q. 
If  now,  we  make 

x"^  =  y,     and  consequently,     x  =:  ±  y  y, 
we  shall  have 


y-  +  2pi/  =  q,     and     y  =  —  p  ±  '\/  q  -\-  p^  : 


hence,  x  =  ±  \/  —  p  d=  y  i 


q  +  p2. 

We  see  that  the  unknown  quantity  has  four  values,  since  each 
of  the  signs  -f-  and  — ,  which  affect  the  first  radical  can  be  com- 
bined in  succession  with  each  of  the  signs  which  aflect  the  sec- 
ond ;  but  these  values  taken  two  and  two  are  numerically  equal,  and 
have  contrary  signs. 

EXAMPLES. 

1,  Take  the  equation 

X*  —  25.t2  =  —  144. 

If  we  make    x"^  =  y,   the   equation  becomes, 
y^  —  25y  =  —  144, 
vvhicli  gives,  y  —  16,     and     y  —  9. 


CHAP.   VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  159 

Substituting  these  values,  in  succession,  for  y  in  the  equation 
a;2  ^^  y^   and  there  will  result, 

1st.     a;2  =  16,     which  gives     a?  =  +  4    and    at  =  —  4. 

2d.      x^  =    9,     which  gives     x  =  -{-  3    and    x  =  —  3. 

Hence,  the  four  values  are    +4,    —  4,    +3,    and    —  3. 

2.  As  a  second  example,  take  the  equation 

X*  —  7a:2  =  8. 
If  we  make  x~  =  y,   the  equation  becomes, 
y2  _  7y  =:  8, 
which  gives  y  r=  8,     and     y  =  —  1. 

Substituting  these  values,  in  succession,  for  y,  and  we  have 
1st.  a;2  =      8,   which  gives   x  =  -{- 2  y      2,    and    x  =  —  2  y      2. 
2d.    a;2  =  — 1,    which  gives    x  = -\-     y— 1,    and    x  =  —     y— 1- 
The  last  two  values  of  x  are  imaginary. 

3.  Let  us  take  the  literal  equation 

X*  —  (2bc  +  4a2)  x-=  —  b^c^. 
By  making   x'^  —     ,  we  have 

y2  _  (25c  +  4a2)  y  =  —  Pc^ ; 

whence,  y  ■=.  he  -\-  2a?  ±  2a  y  ic  -f-  cP- ; 

and  consequently, 

a;  =  ±  \J he  +  2a2  ±  2a  ^/hc  +  a^, 

4.  Suppose  we  have, 

2x  —  lyf~x  =  99. 

If  we  make    yf  x  =  y,   we  have    «  =  y^,    and  hence, 

2y2  —  7y  =  99  ; 

from  which  we  obtain 

^1 
y  =    9,     and     y  = — -. 

121 

hence,  a?  =  81,     and     x  =       , 

4 

153.  Before  resolving  the  general  case  of  trinomial  equation?, 
it  may  be  well  to  remark  that,  the  nth  root  of  any  quantity,  is  /i.-i 
fxpression  which  mitltiplied  hy  itself  n — 1  tiines  will  produce  the 
^ii'€7i  quantity. 


160  ELEMENTS    OF    ALGEBRA.  [CHAP.  VI. 

The  method  of  finding  the  nth  root  has  not  yet  been  explained, 
but  it  is  sufficient  for  our  present  purpose  that  we  are  able  to  in- 
dicate it. 

Let  it  be  required  to  find  the  values  of  y  in  the   equation 

yin  +  2py-  =  q. 
If  we  make   y"  =  x,   we  have   y^n  _  ^p.^    and  hence,  the   given 
r.juation  becomes 

x^  +  Ipx  z=z  q, 

and  hence,  x  =  —  p  ±:  yf  q  +  p"^ ; 

that  is,  y"  =  —  p  zb  y  ^  ■\-  p^. 


and  y  =  \J  —  p  ±i  ■>/ q  +  p"^. 


If  we  suppose  n  =  2,  the  given  equation  becomes  a  trinomial 
equation  of  the  fourth  degree,  and  we  have 

y  =  Y  —  p  ±  y?  +  /• 

154.  The  resolution  of  trinomial  equations  of  the  fourth  degree, 
therefore,  gives  rise  to  a  new  species  of  algebraic  operation  :  viz., 
the  extraction  of  the  square  root  of  a  quantity  of  the  form 

a  ±  y/~b, 
in  which  a  and  b  are  numerical  or  algebraic  quantities. 

To  illustrate  the  transformations  which  may  be  effected  in  ex- 
pressions of  this  form,  let  us  take  the  expression  3  ±  y  5. 

By  squaring  it   we  have 

(3  ±  yT^f  =  9  ±  6  V^  +  5  =  14  ±  6  ^/T: 

hence,  reciprocally,     \/14itG-y/5   =3dry5. 
As  a  second  example,  we  have 

(/?  rh-v/TT)^  =  7  ±  2^77+  11  =  18±2  V^: 

hence,  reciprocally,     \/ \Q  ±2  -J  11  =  -i/T"  ±  -y/Tl- 
Hence  we  see,  that  an  expression  of  the  form 


4^ 


may  sometimes  be  reduced  to  the  form 

a'  ±  y/  1/    or     yHi'  ±  y/17\ 


CHAP.  VI.]  EQUATION'S    OF    THK    SECOND    DEGREE.  161 

and  when  this  transformation  is  possible,  it  is  advantageous  to 
effect  it,  since  in  this  case  we  have  only  to  extract  two  simple 
square  roots  ;    whereas,  the  expression 


ya 


^ecjuires  the  extraction  of  the  square  root  of  the  square  root. 

155.  If  we  represent  two  indeterminate  quantities  by  p  and  y- 
we  can  always  attribute  to  them  such  values  as  to  satisfy  tin' 
(vquations 

p  +  ?  =  \/«  +  /y  -  .  -  (1), 

and  p  —  q  =  \J  a  —  ^fT    -     -     -     (2). 

Xhese  equations  being  multiplied  together,  give 


J»2  —  ^2  _  ^ ^2  —  J      ....      (3). 

Now,  if  p  and  q  are  irrational  monomials  involving  only  single 
radicals  of  the  second  degree,  or,  if  only  one  is  irrational,  it  fol- 
lows that  p2  and  q^  will   be  rational ;    in  which  case,  p-  —  q"^,  or 

its  value,  -y/ a^  —  b,  is  necessarily  a  rational  quantity,  and  conse- 
quently,   a^  —  b   is  a  perfect  square. 

Under  this  supposition,  a  transformation  can  always  be  effectp<l 
that  will  simplify  the  expression. 

By  squaring  equations  (1)  and  (2),  we  have 

p2  4. 2pq  ^q--  =  a^  yy 

p2  _  2pq  -\-  q"^  z=  a  —  -y/Y, 

and  by  adding  member  to  member, 

p"^  -{-  q'^  =  a     -     -     (4). 
If  we  denote  the  second  member  of  equation  (3)  by  c,  wo  shall 
have 

p2  _  ^2  _  g     .     .     ^5^ 

By  adding  the  two  last  equations  and  subtracting  equation  [:'>) 
from  (4),  we  have 

2/)2  =       a  -\-  c,      aTid     2q~  =        a  —  c . 


^    i'^V^— '     ^'"^      ?  =\/^ 


and  therefore,     p  — \/^^— !-- " 


11 


162  ELEMENTS    OF    ALGKIiRA.  [CHAP.   VI 

and  consequently, 

\Ja  +  ^,     or     p+q=±\/"-4r^±\/~-. 


Sja-V^,     or     p-,=  ±\/l  +  i^\r^; 


hence, 


\/^W?=±(\/^^  +  \/M  - 


(6), 


\/WT=:.(\/I±-'-V^O     - 


and      y„_V*  =  d=^V-^--V^-j    -     -    (7). 

These  two  formulas  can  be  verified ;  for  by  squaring  both  mem 
bers  of  the  first,  it  becomes 

r—      a  -\-  c       a  —  c  '     cP-  —  c^  i 

fl  +  /  6  =  — ^  +  -^—  +  2  y  --p-  =  a  +  Va2  -  c2 ; 

but,  sj  cp-  —  b  =  c,     gives     c-  =  a^  —  h. 

Hence,         a  -f  ^fT  —  a  +  ^J  a^  —  d^  +  b  —  a-\-y/h. 
The  second  formula  can  be  verified  in  the  same  manner. 

Remark. — 156.  Formulas  (6)  and  (7)  have  been  deduced  with 
out  reference  to  any  particular  value  of  c ;  and  hence,  they  are 
equally  true  whether  c  be  rational  or  irrational.  If,  however,  c  is 
irrational,  they  will  not  simplify  the  given  expression,  for  each 
will  contain  a  double  radical.  Therefore,  in  general,  this  trans- 
formation is  not  used,  unless    a^  —  b   is  a  perfect  square. 

EXAMPLES. 


1.  Reduce  y   94  +  4J  -/s  =\/  94  +  -y/  8820,    to    its    sim- 

j>lest  form.     We  have  a  =  94,     h  =  8820, 

whence,  c  =  ^^2  -b  =  y/sSZQ  —  8820  =  4, 

a   rational    quantity ;    therefore,   formula    (6)    is    applicable    to   this 
case,  and  we  have 

or,  reducing,  =  ±  (y  49  +  y  45)  ; 


CHAP.  VI.]  EQUATIOXS    OF    THE    SECOND    DEGREE.  163 

y  94  +  42  ^fb  =  ±  (7  +  3  ^T). 


therefore, 
Indeed, 

(7  +  3  y/~bf  =  49  +  45  +  42  ^fb  =  94  +  42  yfb  . 

2.  Reduce  \/  «p  +  S/n^  —  2m  s/ np  +  m^^  to  its  simplest 
form.     We  have 

a=i  np  -\-  2m^,     and     b  =  Am?'  [np  +  m"^), 

d^  —  6  =  ri^p-,     and     c  =  ■\/ a?  —  b  =.  np ; 
and  therefore,  formula  (7)  is  applicable.     It  gives, 

/    /np  +  2m^  -{-  np  np  +  2Tr?  —  np\ 

±iv — ^ V- — ^ /' 

and,  reducing,  ±  {y  np  +  m^  —  »?). 

Indeed,  {ynp  +  »i^  —  w)^  =  «/)  +  2m'^  —  2m  yf  np  +  rr?. 

3.  Reduce  to  its  simplest  form, 

y  16  +  30  y  -  1  +  Y   16  -  30  y  -  1- 
By  applying  the  formulas,  we  find 

y   16  +  30  y  -  1  =5  +  3  y  —  1, 
and  y  16  —  30  ^  —  1  =5  —  3  yj  —  1  : 

hence,         y   16  +  30  v^^  1  +  y    16  —  30  ^  —  I  =  10. 

This  last  example  shows  very  clearly  the  utility  of  the  general 
problem ;  because  it  j)roves  that  imaginary  expressions  combined 
together,  may  produce  real,  and  even  rational  results. 

4.  Reduce  to  its  simplest  form. 


y  28  +  10  V^ .  Ans.  5  +  ^3^ 

5.  Reduce  to  its  simplest  form, 

y   1  +  4  y/^^.  Ans.  2  +  y/~^Z 

6.  Reduce  to  its  simplest  form, 

y  be  +  2b  y/Tc  —  b"^  -r    \J  be  --2h  y/bc  —  Z-2. 


Ans.    rt  2b 


ICri  ELEMENTS    OF    ALGEBRA.  [CHAP.  VI. 

7.  Reduce  to  its  simplest  form, 


V' 


ab  +  4c2  —  (Z2  _  2  y/Aahc^  —  ahcP. 

Ans.  yjab  —  -s/ Ac^  —  tl^. 


Equations  of  the   Second  Degree  involving  two  or  more 
Unknown    Quantities. 

157.  An  equation  involving  two  or  more  unknown  quantities,  is 
said  to  be  of  the  second  degree,  when  the  greatest  sum  of  the  ex- 
ponents (f  the  nnhnown  quantities,  in  any  term,  is  equal  to  2.     Thus, 

3X2      4^      _y     y1      ^y     f/y     -(-      G      =      0 ,  IXIJ      \X     A^      tj     =-     ^ , 

are  equations  of  the  second  degree. 

Hence,  every  general  equation  of  the  second  degree,  involving 
two  unknown  quantities,  may  be  reduced  to  the  form 
a-f  -\-  hxy  +  ca;2  ■\-  dy  -\-  fx  -\-  g  z=z.  (i, 

a,  b,  c,   &c.,  representing   known   quantities,    either   numerical   or 
algebraic. 

Take  the  two  equations 

a  y-  -\-  b  xy  -{-  c  x"^  -{-  d  y  -{-  f  X  -{-  g  =0, 
a'y'^  +  h'xy  +  c'x"^  -\-  d'y  -\-  f'x  -(-  ^  =  0. 
Arranging  them  with  reference  to  x,  they  become 
cx^-\-{by■\■f)x^-ay'^-\-dy■\-g  m  0, 
cV  +  (//y  +/0  .T  +  fl>2  _f.  j/y  +  ^  ^  0 ; 
from  which  we    may   eliminate    or,    after  having   made  its   co-effi- 
cient the  same  in  both  equations. 

By  multiplying  the  first  equation  by  c' ,  and  the  second  by  c, 
they  become 

cc'x2  +  (iy  4-/)c'a:+  (ay^  +  Jy +^  )  </ =  0, 
fcV  ^{h'y  ^f)cx^  [ay  +  d'y  -\- g")  c  =  0. 
Subtracting  one  from  the  other,  we  have 
[(b(/  —  cb') y  -\-fc  —  cf]  x+  {ac'  —  ca')  y"^  +  {d(/—  cd')  y  +  gc' 

-  eg"  =  0, 
which  gives 

_  (cr/  —  ac')  7/2  4-  {cdf  —  dc')  y  -\-  eg'  —  g</ 
{be  -ch')y+fc'-ef  '     • 


CHAP.   VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  165 

This  value,  being  substituted  for  x  in  one  of  the  proposed  equa- 
tions, will  give  a  final  equation^  involving  y. 

But  without  effecting  the  substitution,  Avhich  would  lead  to  a 
very  complicated  result,  it  is  easy  to  perceive  that  the  final  equa- 
tion involving  y  will  be  of  the  fourth  degree.  For,  the  numerator 
of  the  value  of  x  being  of  the  form  wiy^  4-  ny  4-  p,  its  square 
will  be  of  the  fourth  degree,  and  this  square  forms  one  of  the 
parts  of  the  result  of  the  substitution. 

Therefore,  in  general,  tlie  resolution  of  two  equations  of  the  sec- 
ond degree,  involving  two  unkiiown  quantities,  depends  upon  that  of 
an  equation  of  the  fourth  degree,  involving  one  unknown  quantity. 

158.  The  manner  of  resolving  a  general  equation  of  the  fourth 
degree,  not  having  been  yet  explained,  we  cannot  here  give  a 
complete  theory  of  this  subject.  We  will,  however,  indicate  some 
of  the  particular  methods  by  which  equations  of  the  second  de- 
gree involving  two  or  more  imknown  quantities,  may  be  resolved 
by  an   equation  of  the   second  degree  involving  but  one. 

1.  Find  two  numbers  such,  thai  the  sum  of  the  respective  prod- 
ucts of  the  first  multiplied  by  a,  and  the  second  multiplied  by  6, 
shall  be  equal  to  2^ ;  and  the  product  of  the  one  by  the  other 
equal  to  p. 

Let  X  and  y  denote  the  required  numbers,  and  we  have 
ax  -\-  hy  =  2s, 

and,  xy  =  p. 

From  the  first 

2s  —  ax 

y  =  — — ; 

whence,  by  substituting  in  the  second,  and  reducing, 
ax'^  —  2sx  =  —  hp. 

rherefcre,  x  =:  —  ±  —  y  s'^  ~  abp, 

a         a 

and  consequently,     y  =  —  zp  —  y/  s^  —  abp, 

Tliis  problem   is  susceptible  of  two  direct  solutions,  because 
s  "^  y  s"^  —  abp, 


166  ELEMENTS    OF    ALGEBRA.  [CHAP.  VI. 

but  in  orJer  that,  the  roots  may  be  real,  it  is  necessary  that 
5^  >     or     =  ahp. 
Let   a  =  Z>  =  1  ;    the  values  of  x,  and  y,  then  reduce  to 

X  =  s  ±.  y  *^  —  p,   and   y  z=  s  ^  y/ s^  —  p  ; 
whence  we  see,  that  under  this  supposition,  the    two  values  of  x 
are  equal  to  those  of  y,  taken  in  an  inverse  order ;  which  shows, 
that  if 


s  +  ^s^  —  p    represents  the  value  of  at,    s  —  -sj s^  —  p 
will  represent  the  corresponding  value  of  y,  and  reciprocally. 

This  relation  is  explained  by  observing,  that,  under  the  last 
supposition,  the  given  equations  become 

a;  -(-  y  =  25',      and     xy  :=z  p ; 

and  the  question  is  then  reduced  to  finding  two  numbers  of  which 
the  sum  is  2s,  and  their  product  p,  or  in  other  words,  to  divide  a 
number  2s,  into  two  such  parts,  that  their  product  may  he  equal  to 
a  given  number  p. 

2.  Find  four  numbers  in  proportion,  knowing  the  sum  2s  of 
their  extremes,  the  sum  2./  of  the  means,  and  the  sum  4c2  of 
their  si|uarcs. 

Let  II,  X,  y,  z,  denote  the  four  terms  of  the  proportion  ;  the 
equations  of  the  problem  will  be 

]  st  condition,  -         -         -                 u-\-  z  =l1s, 

2d  condition,  -         -         -                 ar  -f  y  =  2^, 

since  they  are  in  proportion,                          uz  z=.  xy, 

4th  condition,  -          -      u"^  -\-  x"^  -\-  y"^  -\-  z"^  =  Ac"^. 

At  first  sight,  it  may  appear  difficult  to  find  the  values  of  tho 
unknown  quantities,  but  with  the  aid  of  an  unknown  auxiliary, 
they  are  easily  determined. 

Let  p  be  the  unknown  product  of  the  extremes  or  meanji- ;  we 
shall  then  have 


and 


Cu  +  z=i2s,')  ...       .  Cu  =  s+^/^—p, 

]                         \  which   give,  ]                      .-- i- 

(       uz  =1  p,     )  Kzz=s  —  ys^—  p. 

(x  +  y=  2s',  }  .  C  -r  =  5^+  ^/s"'  -  p 

]                          [  which  give,  j                            .^ 

V        xy  —  p,      J  \  y  =  s  —  y/  s  ^  —  p. 


CHAP.   VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  167 

Hence,  we  see  that  the  determination  of  the  four  unknown  quan- 
tities depends   only  upon  that  of  the  product  p. 

Now,  by  substituting  these  values  of  w,  x,  y,  z,  in  the  last  of 
the  equations  of  the  problem,  it  becomes 


+  {s'  -  y/s'"^  —  pf  =  46-2  ; 
and  by  developing  and  reducing, 

4i-2  4-  4.y'2  _  4p  —  4^2 ;    hence,     p  z=  s^  -\-  s''^  —  c^. 

Substituting  this  value  for  p,  in  the  expressions  for  u,  x,  y,  z, 
we  find 


^  M  =  6-  +  Y  <::2  —  s''^,  r  X  =:  s'  -\-  ^/  c'^  —  s^-, 

\  Z  =^  S  —  Y  c2  —  ^-'2^  ^   jy  —.  g/  —   y'    c2  —   .«2. 

These  four  numbers  evidently  form  a  proportion ;    for  we  liave 


UZ  =.  [S    +   V  c2  —  s'"^)   {s  —  Y  c2  —  ^'2)   =  ^2     —   c"^  -\-   .v'2, 

xy  =  (5'  +  Y  ^^  —  ^^)  {^ —  V  <^^  —  s^)  =2  s''^  —  c2  4-  -i'  ^• 
Remark. — This  problem  shows  how  much  the  introduction  ot 
an  unknown  auxiliary  facilitates  the  determination  of  the  principal 
unknown  quantities.  There  are  other  problems  of  the  same  kind, 
which  lead  to  equations  of  a  degree  superior  to  the  second,  and 
yet  they  may  be  resolved  by  the  aid  of  equations  of  the  first  and 
second  degrees,  by  introducing  unknown  auxiliaries. 

3.  Given  the  sum  of  two   numbers  equal  to  a,  and  the  sum  of 
their  cubes  equal  to  c,  to  find  the  numbers 

-\-  y    —a 
-|-  y-^  =  c. 
Putting     x  ^:  s  -{-  z,     and     y  z:^  s  —  z,     we  have     a  =  2j, 

2sH  +  35^2  4.  ^3 


By  the  conditions  <     ^ 

z,     and     y 

C  x^  =  .s^  + 

Xy"^  z=  s^  —  ^s^z  +  Zsz"^ 


hence,  by  addition,      x^  +  y^  =  2.y3    +  Qsz"^  =  c ; 


,       c  -  2s^  ,  ^  /c  -  2A-3 

whence,  z^  =  — ,      and      z  =  ±\/  — , 

OS  *        6s 


x  =  s±i\l  — ;     and    y  —  s::i^\j- 


'c  —  2.S-" 
or, 


e>s       '  y--^  y       Q^ 


168  ELEMENTS    OF    ALGEBRA.  [CHAP.  V3 

and  by  substituting  for  s  its  value, 

2  V  V      3a     /         2         V       12a    ' 

and        y^_^V(-^f-)-T-V-T^- 

4.  Given,     ——  =  48,     and     —^  =  24,    to  find  x  and   y. 
Y  a;  Y  a; 

Dividing  the  first  equation  by  the  second,  we   have 
=  Y  y  =  2,     and    hence    y  =  4. 


Whence,  from  the   second  equation  we  have, 

4,T 


4^^  =24, 

Y  a; 

and  consequently,        y  a:  =  6,     and     x  =  36. 

5.  Given,     x   +  V  •^'Z  +  V    =     19  )  ^    ,  , 

_  o  o  >       to  find  X  and  y. 

and  a;'  +       a:;/  +  y^  =  133  ) 

Dividing  the  second  equation  by  the  first,  we  have 
X  —  y/Ty  -\-y  =    1, 

but,  X  4-  -s/xy  +  y  =  19: 

hence,  2a;  +  2y  =  26    by  addition, 

or,  a;  +    y  =  1 3  ; 

and  y^y  +  13  =  19    by  substituting  in  the  1st  eq. ; 

or,  -^xy  =    6 

and  a;y  =  36. 

The  2d  equation  is,     x"^  -\-    xy  -\-  y"^  ^  133, 
and  from  the  last,  3a'y  =108; 

by  subtracting  a^  —  2a'y  +  y^  =    25  : 

hence,  x  —  y  ^=  zt:      5. 

But,  X  -\-  y  =z  13 : 

hence,  at  =  9,    or    4  ;     and     y  =  4,    or    9. 


CHAP.   VI.]  EQUATIONS    OF    THE    SECOND    DEGREE.  169 

6.  Find  the  values  of  cc  and  y,  in  the  equations 
a;2  +  3a:  +  y  =  73  —  2x1/ 
1/"^  -\-  3y  -\-  X  =  44. 
By  transposition,  the  first  equation  becomes, 
a;2  +  2xy  +  3a;  +  y  =  73  ; 
to  which,  if  the  second  be  added,  there  results, 

a:2  +  2a-y  +  y2  +  4t  +  4y  =  (a-  +  y)2  +  4  (a:  +  y)  =  117 
[f  now,  in  the  equation 

we  regard  a:  +  y   as  a  single  unknown  quantity,  we  shall  have 
x  +  y=  —2  ±1  yil7+  4; 
hence,  a;  +  y=— 2  +  11  =  9, 

and  a;  +  y=— 2  —  11  =  —  13; 

whence,  a?  =  9  —  y,    and    x  =  —  13  —  y. 

Substituting  these  values  of  x  in  the  second  equation,  we  have 

y2  4-  2y  =  35,     for     x  =  9  —  y, 
and  y2  4-  2y  =  57,     for     a?  =  —  i3  —  y. 

The  first  equation  gives, 

y  =  5,     and     y  =  —  7. 
and  the  second, 

y  =  —  1  +  -/Hi'     and     y  =  —  1  —  -/58. 
The  corresponding  values  of  x,  are 

a;  =  4,         0?  =  16  ; 
a;  =  _  12  —  ^58,     and     a:  =  —  12  +  -/sS 
7'  Find  the  values  of  x  and  y,  in  the  equations 
x'^y'^  +  ^y^  -\-  xy  =^  600  —  (y  +  2)  x'^y^ 
X  -{-  y'^  =  14  —  y. 
From  the  first  equation,  we  have 

a;2y2  -f  (y2  -|-  2y)  ar^y^  -|-  xy"^  -f  xy  =  600, 

or,  xY  (1  +  y^  +  2y)  +  a:y  (1  +  y)  =  600, 

or,  again,      x'^y'^  (1  +  y)^  -|-  a:v  (1  +  y)  =  600  ; 


170  ELEMENTS    OF    ALGEBRA.  ICHAP.  VI 

which  is   the   form  of  an   equation  of  the    second    degree,  by  re- 
garding   a?y(l  +y)    as  the  unknown  quantity.     Hence, 


/ /2 

(1  +  y)  =  -  A  db  V600  +i=-i±\/- 


'2401 


4     ' 

and  if  we  discuss    only  the    roots  which   belong    to   the   +  value 
of  the  radical,  we  have 

49 

^y(i4-y)=-i  +  -2-=24; 

and  hence,  o 


y  +  y^ 

Substituting  this  value  of  x  in  the  second  equation,  we  liave 

{y'  +  yy-U(y'  +  y)  = -24; 

whence,  y2  _|_  y  __  12,     and     y^  _j_  y  _  2.  . 

From  the  first  equation,  we  have 

y  = ±  —  =  3,    or     — 4; 

and  the  corresponding  values  of  x,  from  the  equation 

24 


f  +  y 
From  the  second  equation,  we  have 

y  =  1,     and     y  =  —  2  ; 
which  gives  a:  =  12. 

8.  Given,     x-y  +  xy"^  =  6,     and     x^y^  +  x^y^  =  12,     to    find   J 
and  y 

Ans. 


x=2     or     1, 
y  =  1     or     2. 


^    ^.  f  a:2 +  a:  +  y  =  18-y2  > 

9.   Given,       <  -^  -^    ^      to  find  a;  and  y. 

(^  xy  =    0  ) 

f  X  =  3,    or    2  ;    or     -3±  y/~2. 

Alls.     <  , — 

C  y  =  2,    or    3  ;    or     —  3  =f  V  3. 


QUESTIONS. 


1.  There  are  two  numbers  whose  difference  is  15,  and  lial!' 
their  product  is  equal  to  the  cube  of  the  lesser  number.  Vvhai 
are  the  numbers?  Ans.  3  and   18 


CHAP.  VI. 1  EQUATIONS    OF    THK    SEnOND    DEGREE.  171 

2.  What  two  numbers  are  those  whose  sum  multiplied  by  the 
greater,  is  equal  to  77 ;  and  whose  difference,  multiplied  by  the 
lesser,  is  equal  to  12  ? 

Ans.  4  and  7,    or    §  v2    and    y  y'2. 

3.  To  divide  100  into  two  such  parts,  that  the  sum  of  their 
square  roots  may  be   14.  Ans.  64  and  36. 

4.  It  is  required  to  divide  the  number  24  into  two  such  parts, 
that  their  product  may  be  equal  to  35  times  their  difference. 

Ans.   10  and   14. 

5.  The  sum  of  two  numbers  is  8,  and  the  sum  of  their  cubes 
is  152.     What  are  the  numbers  ?  Ans.  3  and  5. 

6.  The  sum  of  two  numbers  is  7,  and  the  sum  of  their  4th 
powers  is  641.     What  are  the  numbers?  Ans.  2  and  5. 

7.  The  sum  of  two  numbers  is  6,  and  the  sum  of  their  5th 
powers  is  1056.     What  are  the  numbers  ?  Ans,  2  and  4. 

8.  Two  merchants  each  sold  the  same  kind  of  stuff:  the  sec- 
ond sold  3  yards  more  of  it  than  the  first,  and  together,  they  re- 
ceived 35  dollars.  The  first  said  to  the  second,  "  I  would  have 
received  24  dollars  for  your  stuff."  The  other  replied,  "  And  I 
would  have  received  12^  dollars  for  yours."  How  many  yards 
did  each  of  them  sell  ? 

C  1st  merchant   x  =:  15  }  fa:=5 

(2d     -     -     -     y=  IS)  (y=8. 

9.  A  widow  possessed  13,000  dollars,  which  she  divided  into 
two  parts,  and  placed  them  at  interest,  in  such  a  manner,  that  the 
incomes  from  them  were  equal.  If  she  had  put  out  the  first  por- 
tion at  the  same  rate  as  the  second,  she  would  have  drawn  for 
this  part  360  dollars  interest ;  and  if  she  had  placed  the  second 
out  at  the  same  rate  as  the  first,  she  would  have  drawn  for  it 
490  dollars  interest.     What  were  the  two  rates  of  interest  ? 

Ans.  7  and  6  per  cent 


172  ELEMENTS    OF    ALGEBRA.  [CHAP.  VII. 


CHAPTER  VII. 

OF    PROPORTIONS    AND    PROGRESSIONS. 

159.  Two  quantities  of  the  same  kind  may  be  compared  lo- 
gether  in  two  ways  : — 

1st.  By  considering  how  much  one  is  greater  or  less  than  the 
other,  which  is  shown  by  their  difference ;    and 

2d.  By  considering  how  many  times  one  is  greater  or  less  than 
the  other,  wliich  is  shown  by  their  quotient. 

Thus,  in  comparing  the  numbers  3  and  12  together  with  re- 
spect to  their  difference,  we  find  that  12  exceeds  3,  by  9  ;  and  in 
comparing  them  together  with  respect  to  their  quotient,  we  find 
that  12  contains  3,  four  times,  or  that  12  is  4  times  as  great  as  3. 

The  first  of  these  methods  of  comparison  is  called  Arithmetical 
Proportion ;   and  the  second.   Geometrical  Proportion.     Hence, 

Arithmetical  Proportion  considers  the  relation  of  quantities 
to  each  other,  with  respect  to  their  difference;  and  Geometrical 
Proportion,  the  relation  of  quantities  to  each  other,  with  respect  to 
their  quotient. 

Of  Arithmetical  Proportion. 

160.  If  we  have  four  numbers, 

2,    4,    8,    and    10, 

of  which  the  difference  between  the  first  and  second  is  equal  to 
the  difference  between  the  third  and  fourth,  these  numbers  are 
said  to  be  in  arithmetical  proportion.  The  first  term  2  is  called 
an  antecedent,  and  the  second  term  4,  with  which  it  is  compared, 
a  consequent.  The  number  8  is  also  called  an  antecedent,  and 
the    number  10,  with  which    it    is    compared,  a  consequent.     The 


CHAP.  VII. 1  ARITHMETICAL    PROGRESSION'.  173 

fir.si  and  fourth   terms    are    called    the    extremes ;    and    the    second 
and  third  terms,  the  means. 

Let  a,  b,  c,  and  e,  denote  four  quantities  in  arithmetical  pro- 
portion ;  and  d  the  difference  between  either  antecedent  and  its 
consequent. 

Then,  a  —  b  =  d,    and    a  z=  b  -\-  d  ; 

also,  c  —  e  ^  d,    and     c  =  c  —  d. 

By  adding  the  last  two  equations,  we  have 

a  -\-  e  ^:z  b  -\-  c  :    that  is, 

If  four  quantities  are  in  arithmetical  proportion,  the  sum  of  the 
two  extremes  is  equal  to  the  sum  of  the  two  means 

Arithmetical  Progression. 

161.  ^\Tien  the  difference  between  the  first  antecedent  and  con- 
sequent is  the  same  as  between  any  two  consecutive  terms  of  the 
proportion,  the  proportion  is  called  an  arithmetical  progression. 
Hence,  an  arithmetical  progression,  or  a  progression  by  differences, 
is  a  succession  of  terms,  each  of  which  is  greater  or  less  than 
the  one  that  precedes  it  by  a  constant  quantity,  which  is  called 
the  common  difference  of  the  progression.     Thus, 

1,     4,     7,  10,  13,  16,  19,  22,  25,  .  .  . 

and  GO,  56,  52,  48,  44,  40,  36,  32,  28,  .  . 

are  arithmetical  progressions.  The  first  is  called  an  increasing 
progression,  of  which  the  common  difference  is  3  ;  and  the  sec- 
ond, a  decreasing  progression,  of  which  the  common  difference  is  4. 

An  arithmetical  progression,  is  also  called,  an  arithmetical  series ; 
and  generally, 

A  series  is  a  succession  of  terms  derived  from  each  other  accord- 
ing to  some  fixed  and  known  law. 

Let  a,  b,  c,  d,  e,  f  .  .  .  designate  the  terms  of  a  progression 
by  differences  ;    it  has  been  agreed  to  write  them  thus  : 

a.b.c.d.e.f.g.h.i.k  .   .  . 

This  series  is  read,  a  is  to  b,  as  b  is  to  c,  as  c  is  to  d,  as  d 
is  to  e,  &c.     This  is  a  series  of  continued  equi-differences,  in  Avliich 


174  ELEMENTS    OF    ALGEBRA.  [CHAP.  VU. 

each  term  is  ut  the  same  time  a  consequent  and  antecedent,  with 
the  exception  of  the  first  term,  which  is  only  an  antecedent,  and 
the  last,  which  is  only  a  consequent. 

162.  Let  d  represent  the  common  diflerence  of  the   progression 
a  .  h  .  c  .  e  .  f .  g  .  h  .  k,  &c., 

which  we  will  consider  increasing. 

From  the  definition  of  a  progression,  it  follows  that, 

b  =  a  -{-  d,     czzzb-[-d=za-\-  2d,      e  =  c  -f  f^  =  «  +  3  J ; 

and,  in  general,  any  term  of  the  series,  is  equal  to  the  first  term 
plus  as  many  times  the  common  difference  as  there  are  precedino 
terms. 

Thus,  let  I  be  any  term,  and  n  the  number  which  marks  the 
place  of  it.  Then,  the  number  of  preceding  terms  will  be  deno- 
ted by  n —  1,  and  the  expression  for  this  general  term,  will  be 

/  =  a  +  («  —  l)rl 

That  is,  any  term  is  equal  to  the  first  term,  plus  the  product  of 
the  common  difference  by  the  numhcr  of  preceding  terms. 

If  we  make  n  =:  1 ,  we  have  I  =  a  ;  that  is,  the  series  will 
have  but  one  term. 

If  we  make 

n  =  2,    we  have    I  =:  a  -]-  d; 

that  is,  the  series  will  have  two  terms,  and  the  second  term  is 
equal  to  the  first  plus  the  common  diflerence. 


EXAMPLES. 

1.  If    a  =  3  and  d  —2,  what  is  the  3d  term?  Ans.  7. 

2.  If    a=  5  and  d  —  4,  what  is  the  Cth  term?  Ans.  25. 

3.  If    a  =  7  and  (1  =  5,  what  is  the  9th  term?  Ans.  47. 
The  formula, 

1  =  a  -\-{n  —  \)d, 

serves  to    find   any  term  whatever,  without    determining   all    those 
which  precede  it. 


CHAP.   VII.]  ARITHMETICAL    PROGRESSION.  175 

Thus,  to  find  the  50th  term   of   the  progression, 

1  .  4  .  7  .  10  .  13  .  16  .  19,  .  .  . 
we  have,  Z  =  1  +  49  x  3  =  148. 

And  for  the  60th  term  of  the  progression, 

1  .  5  .  9  .  13  .  17  .  21  .  25,  .  .  . 
Ave  have,  Z  =  1  +  59  x  4  =  237. 

163.  If  the  progression  were  a  decreasing  one,  we  should  have 

I—  a  —  {n  —  l)d. 

That  is,  any  term  in  a  decreasing  arithmetical  progression,  is  equal 
to  the  jirst  term  minus  the  product  of  the  common  difference  hy  the 
number  of  preceding  terms. 

EXAMPLES. 

1.  The  first  term  of  a  decreasing  progression  is  60,  and  the 
common  difference  3  :  what  is  the  20lh  term  ? 

1  =  a  —  {n  —  \)d    gives     /  =  60  —  (20  —  1)  3  =  60  —  57  =  3. 

2.  The  first  term  is  90,  the  common  difiference  4  :  what  is  the 
15th  term  ?  Ans.  34. 

3.  The  first  term  is  100,  and  the  common  diflference  2  :  what 
is  the  40th  term  ?  Ans.  22. 

164.  A  progression  by  differences  being  given,  it  is  proposed 
to  prove  that,  the  sum  of  any  two  terms,  taken  at  equal  distances 
from  the  two  extremes,  is  equal  to  the  sum  of  the  two  extremes. 

Let  a  .  b  .  c  .  e  .  f  .  .  .  .  i  .  k  .  I,  be  the  proposed  progression, 
and  n  the  number  of  terms. 

We  will  first  observe  that,  if  x  denote  a  term  wdiich  has  p 
terms  before  it,  reckoning  from  the  first  term,  and  y  a  term  which 
has  p  terms  before  it,  reckoning  from  the  last  term,  we  have, 
from  what  has  been  said, 

X  =  a  +  p  X  rf, 

and  y=:l  —  pxd; 

whence,  by  addition,      x  -\-  y  —  a  -\-  I. 


176  ELEMENTS    OF    ALGEBRA.  [CHAP.  VII. 

Now,  to  find  the  sum  of  all  the  terms,  write  the  progression 
below  itself,  but  in  an  inverse  order,  viz., 

a  .  h  .  c  .  e  .f  .  .  .  .  i  .  k  .  I. 

I  .  k  .  i c  .  b  .  a. 

Calling  S  the  sum  of  the  terms  of  the  first  progression,  2>S 
will  be  the  sum  of  the  terms  in  both  progressions,  and  we  sli;i!l 
have 

2S^{a-{-J)-{-{b  +  k)  +  {c  +  i)  .  .  .  +{i  +  c)  +  {k  +  h)  +  {l  +  a). 

And,  since  all  the  parts  a  -j-  I,  b  -{-  k,  c  -\-  i  .  .  .  .  are  equal 
to  each  other,  and  their  number  equal  to  n,  by  which  we  desig- 
nate the  number  of  terms  in   each  series,  we  have 


2S 


-\-h 


(a  +  /)n,     or      S  =  {^-^) 


That  is,  the  sum  of  the  terms  of  an  arithmetical  progression,  is 
equal  to  half  the  sum  of  the  two  extremes  multiplied  by  the  number 
of  terms. 


EXAMPLES. 


1.  The    extremes    are  2   and   IG,  and   the    number  of  terms  8 
what  is  the  sum  of  the  series  ? 


S 


/a  +  h  .  „       2  +  16 

[-^—)  X  n,     gives     S  =  — ^ X  8  =  72. 


2.  The  extremes  are  3   and  27,  and  the  number  of  terms  12  : 
what  is  the  sum  of  the  series?  Ans.  180. 

3.  The   extremes  are   4   and  20,   and  the  number  of  terms  10: 
what  is  the  sum  of  the  series  ?  Ans.  120. 

4.  The  extremes  are  8   and   80,   and  the  number  of  terms  10: 
what  is  the  sum  of  the  series  ''  A?is.  440. 

165.  The  formulas 

fa+  h 
~2 


I  =  a  -\-  [n  —  \)  d     and     S  =  ( ]  X  n. 


contain  five  quantities,  a,  d,  n,  I,  and  S,  and  consequcmlly  give 
rise  to  the  following  general  problem,  viz. :  Any  three  of  these 
five  quantities  being  given,  to  determine  the  other  two. 


CHAP.  VII.]  ARITHMETICAL    PROGRESSION.  177 

This  general  problem  gives  rise  to  the  ten  following  cases : — 


No. 


Given. 


c,  d,  n 


a,  d,  I 


a,  d,  S 


a,  n,  I 


a,  n,  S 


a,  I,  S 


d,  n,  I 


d,  n,  S 


d,  1,  S 
n,l,S 


I,  S 


n,   S 


S,  d 


d,  I 


n,  d 


a,   S 


a,  I 


a,  d 


Values  of  the  unknown  quantities 


1=  a-{-  {n—l)d;   S  —  ^n  [2a -\-  (ti  —  1)  d]. 


l-a                        (Z  +  a)  (/  -  a  +  J) 
-^+^'     -^^^ 2d • 


d—2a±  J(d-2ay  +  8dS     , 


)d. 


S=ln(a  +  l);     d  = 


l-a 
n  —  1' 


2(-S  — on)      j_2S 
n  [n  —  1)  '  71 


2S  ,        (l-^a)(l  —  a) 

»  =  — : — 7 ;    d  = 


a  +  I 


2S  —  (l-\-  a) 


a=zl -{n~l)d;     S  =  ^n  [21  -  (?i  -  1)  d].    i 


2S  —  n(n  —  l)d        ,        2S  +  n{n  —  l)d 
a  = :: ;     /  =  — 


2n 


2n 


2/4- J±  ^/{2l-j-d)^^-8dS 


2d 


;  a  =  l  —  (ra  —  l)d. 


2S        ,       ^       2(7iJ-S) 

a  = /  ;    d  =  — — . 

n  n  [n  —  1) 


The  solution  of  these  cases  presents  no  difficulty.  Cases  3  and 
9  give  rise  to  equations  of  the  second  degree  ;  but  one  of  the 
roots  will  always  satisfy  the  enunciation  of  the  question  in  its 
arithmetical  sense. 

If  we  resume  the  formula 

I  ^  a  -{-  {n  —  I)  d, 

we  have,  a  z=  I  —  (n  —  \)  d\    that  is, 

The  first  term  of  an  increasing  arithmetical  progression,  is  equal 
to  any  following  term,  minus  the  product  of  the  common  differc7ice 
by  the  number  of  preceding  terms. 

From  the  same  formula,  we  also  find 

l-a 


d  = 


n  —  1 


;    that  is, 


12 


178  ELEMENTS    OF    ALGEBRA.  [CHAP.    VIl 

In  any  arithmetical  progression,  the  common  difference  is  equal  to 
the  difference  between  the  first  and  last  terms  considered,  divided  by 
the  ?inmber  off  terms  less  one. 

1.  Two  terms  of  a  progression  are  16  and  4,  and  the  number 
of  terms  considered  is  5  :    what  is  the  common  difference  ? 

The  formula 

l~a       .  ,       16-4       ^ 

a  = gives     d  = =  3. 

n  —  1      *=  4 

2.  Two  terms  of  a  progression  are  22  and  4,  and  the  number 
of  terms  considered  is   10  :    what  is  the  common  difference  ? 

Ans.  2. 

166.  The  last  principle  affords  a  solution  to  the  following 
question  : — 

To  ffnd  a  number  m  of  arithmetical  means  between  two  given 
numbers  a  and  b. 

To  resolve  this  question,  it  is  first  necessary  to  find  the  com- 
mon difference.  Now  we  may  regard  a  as  the  first  term  of  an 
arithmetical  progression,  6  as  a  subsequent  term,  and  the  required 
means  as  intermediate  terms.  The  number  of  terms  of  this  pro- 
gression which  are  considered,  will  be  expressed  by   m  -\-  2. 

Now,  by  substituting  in  the  above  formula,  b  for  /,  and  m  -\-  2 
for  n,  it  becomes 

b  —  a  ,        h  —  a 

d  = -,     or 


m  +  2  —  \  m  +  1 

that  is,  the  common  difference  of  the  required  progression  is  ob- 
tained by  dividing  the  difference  between  the  given  numbers  a 
and  b,  by  one  more  than  the  required  number  of  means. 

Having  obtained  the    common    difference,  form  the  second  term 
of  the  progression,  or  the  first  arithmetical  mean,  by   adding  d,  or 

,    to  the  first  term  a.     The  second  mean  is  obtained  by  aug- 

//i  -|-  1 

menting  the  first  by  d,  &c. 

1.  Find  3  arithmetical  means  between  2  and  18.     The  formula 

b  -a         .            ,       18-2       ^ 
d  = ,    gives     a  = =  4  ; 

nencn  the  progression  is 

2  .  6  .  10  .  14  .  18. 


CHAP.  VII.]  ARITHMETICAL    PROGRESSION.  179 

2.  Find  12  arithmetical  means  between  12  and  77.  The  for- 
mula 

b  —  a         .  ,       77  —  12 

d  = -— ,    gives    d  = — =  5  ; 

m  -\-  I  Id 

hence  the  progression  is 

12  .  17  .  22  .  27 72  .  77. 

167.  Remark. — If  the  same  number  of  arithmetical  means  are 
inserted  between  the  terms  of  a  progression,  taken  two  and  two, 
these  terms,  and  the  arithmetical  means  united,  will  form  one  and 
the  same  progression. 

For,  let  a  .  6  .  c  .  e  ./  .  .  .  .  be  the  proposed  progression,  and 
m  the  number  of  means  to  be  inserted  between  a  and  b,  b  and  c, 
c  and  e 

From  what  has  just  been   said,  the   common  difference   of  each 
partial  progression  will  be  expressed  by 
h  —  a         c  —  b         e  —  c 
m-f-l'      m-j-l'      m  -\-  \ 
which    are    equal  to    each   other,   since    a,  b,  c,  .  .  .  are    in    pro- 
gression :    therefore,  the   common  difference   is  the  same  in   each 
of  the  partial  progressions  ;   and  since  the   last   term  of  the    first, 
forms  the  first  term  of  the   second,   &c.,   we    may   conclude    that 
all  of  these   partial  progressions  form  a   single  progression. 

EXAMPLES. 

1.  Find  the  sum  of  the  first  fifty  terms  of  the  progression 

2  .  9  .  16  .  23  .  .  . 

For  the  50th  term,  Ave  have 

Z  =  2  4-  49  X  7  =  345. 

50 
Hence,         -S  =  (2  +  345)  x  —  =  347  X  25  =  8675. 

2.  Find  the   100th  term   of  the  series    2  .  9  .  16  .  23  .  .  . 

A71S.  695. 

3.  Find  the  sum  of  100  terms  of  the  series    1  .3.5.7.9... 

Ans.   10000 

4.  The  greatest  term  considered  is  70,  the  common  difference 
3,  and  the  number  of  terms  21  :  what  is  the  least  term  and  the 
sum  of  the   series  ? 

Ans.  Least  term  10  ;    sum  of  series  840. 


180  ELEMENTS    OF    ALGEBRA.  [CHAP.   VII 

5.  The  first  term  of  a  decreasing  arithmetical  progression  i? 
10,  the  common  difference  one  third,  and  the  number  of  terms 
21  :  reqni-red  the  sum  of  the  series.  Ans.  140. 

6.  In  a  progression  by  differences,  having  given  the  common 
difference  6,  the  last  term  185,  and  the  sura  of  the  terms  2945 . 
find  the  first  term,   and  the  number  of  terms. 

Ans.  First  term    =  5  ;    number  of  terms   31. 

7.  Find  9  arithmetical  means  between  each  antecedent  and  con. 
sequent  of  the  progression   2. 5.  8.  11.  14... 

Ans.  d  =  0.3. 

8.  Find  the  number  of  men  contained  in  a  triangular  battalion, 
the  first  rank  containing  1  man,  the  second  2,  the  third  3,  and 
so  on  to  the  n"',  which  contains  n.  In  other  words,  find  the  ex- 
pression for  the   sum  of  the   natural  numbers    1,  2,  3,  .  .  .  from 

1    to   n,  inclusively.  .         ^ n(ra+l) 

o 

9.  Find  the  sum  of  the  n  first  terms  of  the  progression  of  un- 
even numbers    1,  3,   5,  7,   9  .  .  .  Ans.  S  =  n^. 

10.  One  hundred  stones  being  placed  on  the  ground,  in  a 
straight  line,  at  the  distance  of  2  yards  from  each  other,  how 
far  will  a  person  travel,  who  shall  bring  them  one  by  one  to  a 
basket,   placed   at  two   yards   from  the   first  stone  ? 

Ans.   11   miles,   840  yards. 

Geometrical  Pro])orlion. 

168.  Ratio  is  the  quotient  arising  from  dividing  one  quantity 
by  another  quantity  of  the  same  kind.  Thus,  if  A  and  B  repre- 
sent quantities  of  the  same  kind,  the  ratio  of  yl  io  B  is  expressed 

B 
A' 

169.  If  there  be  four  magnitudes,  A,  B,  C,  and  D,  having  such 
values  that 

B  _D^ 
A"  C 

then  A   is   said    to  have   the  same  ratio  to   B,  that   C  has  to  I)  \ 
or,  the  ratio  o^  A  to  B  is  equal   to    the  ratio  of  C  to  D.     Wlien 


CHAP.   VII.]  GEOMETRICAL    PROPORTION.  181 

four  quantities  have  this  relation  to    each   other,  they  are   said    to 
be  in  proportion.     Hence,  proportion  is  an  equality  of  ratios. 

To  express  that  the  ratio  of  ^  to  5  is  equal  to  the  ratio  of  C 
to  D,  we  write  the  quantities  thus, 

A  :  B  :   :   C  :  D, 
and  read,  A  is  to  B,  as   C  is  to  D. 

The  quantities  which  are  compared  together  are  called  the  terms 
of  the  proportion.  The  first  and  last  terms  are  called  the  two  ex- 
tremes, and  the  second  and  third  terms,  the  two  means. 

170.  Of  four  proportional  quantities,  the  first  and  third  are  called 
the  antecedents,  and  the  second  and  fourth  the  consequents;  and 
the  last  is  said  to  be  a  fourth  proportional  to  the  other  three 
taken  in  order. 

171.  Three  quantities  are  in  proportion  when  the  first  has  the 
same  ratio  to  the  second  that  the  second  has  to  the  third  ;  and 
then  the  middle  term  is  said  to  be  a  mean  proportional  between 
the  other  two, 

172.  Quantities  are  said  to  be  in  proportion  by  inversion,  or  in- 
versely, when  the  consequents  are  made  the  antecedents  and  the 
antecedents  the  consequents. 

173.  Quantities  are  said  to  be  in  proportion  by  alternation,  or 
alternately,  when  antecedent  is  compared  with  antecedent  and  con- 
sequent with  consequent. 

174.  Quantities  are  said  to  be  in  proportion  by  composition,  when 
the  sum  of  the  antecedent  and  consequent  is  compared  either  with 
antecedent  or  consequent. 

175.  Quantities  are  said  to  be  in  proportion  by  division,  when 
the  difference  of  the  antecedent  and  consequent  is  compared  either 
with  antecedent  or  consequent. 

176.  Equi-multiples  of  two  or  more  quantities  are  the  products 
which  arise  from  multiplying  the  quantities  by  the  same  number. 
Thus,  m  X  A  and  m  x  B,  are  equi-multiples  of  A  and  B,  the 
common  multiplier  being  m. 

177.  Two  quantities,  A  and  B,  are  said  to  be  reciprocally  pro- 
portional, or  inversely  proportional,  when  one  increases  in  the  same 
ratio  as  the  other  diminishes.  When  this  relation  exists,  eilher 
of  them  is  equal  to   a  constant  quantity  divided  by  the   other 


182  ELEMENTS    OF    ALGEBRA.  [CHAP.  VII 

178.  If  we  have  the  proportion 

A  :  B  :  :   C  :  D, 

7?         T) 

we  have  —  =  — -,    (Art.   169); 

and  by  clearing  the   equation  of  fractions,  we  have 

BC  =  AD  ;    that  is, 

Of  four  proportional  quantities,  the  product  of  the  two  extremes 
is  equal  to  the  product  of  the  two  means. 

179.  If  four  quantities,  A,  B,  C,  and  D,  are  so  related  to  each 
other  that 

A  X  D  =  B  X  C, 

we  shall  also  have,  ^  =  ^. 

A         O 

and  hence,  A  :  B  :   :   C  :  D ;    that  is 

If  the  product  of  two  quantities  is  equal  to  the  product  of  two 
other  quantities,  two  of  them  may  he  made  the  extremes,  and  the 
other  two  the  means  of  a  proportion. 

180.  If  we  have   three   proportional  quantities, 

A  :  B  :  :  B  :   C, 

we  have  —-=:—-; 

A        B 

hence,  B"^  z=z  AC ;    that  is, 

The  square  of  the  middle  term  is  equal  to  the  product  of  the  two 
extremes. 

181.  If  we  have 

A  :  B  :  :   C  :  D,     and  consequently,    —  =  — , 

jri.  O 

multiplying  both  members  of  the  equation  by    -^,    we  obtain 

JO 

C       D^ 
1-  B' 

and  hence,  A  :   C  :   :  B  :  D  ;    that  is. 

If  four  quantities  are  proportional,  they  will  be  in  proportion  bij 
alternation. 


D 

B       F 

and 

c" 

'a~e 

CHAP.  VII. J  GEOMETRICAL    PROPORTION.  183 

182.  If  we  have 

A  :  B  :   :    C  :   D,    and    A  :  B  :  :  E  :  F, 

we  shall  also  have 

B 
'A 

D       F 

hence,         tt  =  ^7     and     C  :  D  :  :  E  :  F ;»  that  is, 
O         E 

If  there  are  two  sets  of  proportions  having  an  antecedent  and  con 
sequent   in    the   one    equal    to    an    antecedent    and    consequent   of  the 
other,  the  remainitig  terms  will  be  proportional. 

183.  If  we  have 

A  :  B  :   :    C  :  D,    and  consequently,    —  =  — , 

"A        G 

we  have,  by  dividing  1  by  each  member  of  the  equation, 

A         C 

-—  =  —-,    and  consequently,    B  :  A  :   :  D  :   C ;    that  is, 
B        D 

Four  proportional  quantities  will  be  in  proportion,  wheji  taken  in- 
versely (Art.   172). 

184.  The  proportion 

A  :  B  :  :   C  :  D,     gives     A  x  D  =  B  x  C. 
To  each  member  of  the   last    equation    add  B  x  D.      We  shall 
then   have 

(A  +  B)  X  D  =  {C+  D)  X  B; 

and  by  separating  the  factors,  we  obtain 

A-\-B  :  B  :  :    C -\- D  :  D. 
If,  instead  of  adding,  we    subtract   B  X  D   from  both  members, 
we  have 

{A-B)  X  D  =  {C-D)xB; 

which  gives      A  —  B  :  B  :   :    C  —  D  :  D ;    that  is, 

If  four  quantities  are  proportional,  they   will  be  in  proportion  hy 
composition  or  division. 

185.  If  we  have 

B__  D_ 
A~  C 


184  ELEMENTS    OF    ALGEBRA.  [CHAP.   Vll. 

and  multiply  the   numerator  and  denominator  of  the  first  member 
by  any  number  m,  we  obtain 

=  -—     and     mA  :  mB  :   :   C  :  D ;    that  is, 

mA         C 

Equal  multiples  of  two  quantities  have  the  same  ratio  as  the  quan- 

t.Lties  themselves. 

186.  The  proportions 

A  :  B  :  :   C  :  D,     and     A  :  B  :  :  E  '.  F. 

give  A  X  D  =  B  X  C,     and     A  x  F=  B  x  E; 

adding  and  subtracting  these  equations,  we  obtain 

A{D±F)=B{C:hE),    or    A  :  B  :  :  C±  E  :  D±F\    that  is. 

If  C  and  D,  the  antecedent  and  consequent,  he  augmented  or 
diminished  by  quantities  E  and  F,  lohich  have  the  same  ratio  as  C 
to  D,  tlie   resulting  quantities  will  also  have   the  same  ratio. 

187.  If  we   have   several  proportions, 

A  :  B  :  :   C  :  D,     which  gives     A  X  D  =  B  X  C, 
A  :  B  :  :  E  :  F,         "  "        A  x  F  =  B  x  E, 

A  :  B  :  :   G  :  H,        "  «        Ax  H=  B  x  G. 

&c.,  &c., 
we  shall  have,  by  addition, 

A{D-{-  F+  H)=B{C  -}-  E  +  G); 
and  by  separating  the  factors, 

A  :  B  :   C  +  E+G:  D  +  F-j-H;    that  is. 
In   any  riumber  of  proportions  having  the  same  ratio,  any  antece- 
dent will  be  to  its  consequent,  as  the  sum  of  the  antecedents  to  the 
sum  of  the  consequents. 

188.  If  we  have  four  proportional  quantities 

A  :  B  :  :   C  :  D,    we  have    —  =  — ■ ; 

and   raising  both  members   to   any  power,   as  the  nth,  we  have 

5"  _  D» 

~A^  ~~C^' 
auii  consequently,  A'^  :  B"  :   :    C"  :   D"  ;    that  is, 


E  :  F  :  :   G  :  H, 


CHAP.  VII.]  GEOMETRICAL    PROGRESSION.  185 

If  four  quantities  are  proportional,  any  like  powers  or  roots  will 
be  proportional. 

189,  Let  there  be  two  sets  of  proportions, 

A  :  B  :  :   C  :  D,    which  gives    —  =  — , 

^  AC 

E       G' 

Muhiply  them  together,  member  by  member,  we  have 

7?P         T)fT 

J^=  QQ^    ^'hich  gives    AE  :  BF  :  :   CG  :  DH;    that  is, 

In  two  sets  of  proportional  quantities,  the  products  of  the  corres- 
ponding terms  will  be  proportional. 


Of  Geometrical  Progression. 

190.  In  the  proportions  which  have  been  considered,  it  has 
only  been  required  that  the  ratio  of  the  first  term  to  the  second 
should  be  the  same  as  that  of  the  third  to  the  fourth.  If  we  im- 
pose the  farther  condition,  that  the  ratio  of  the  second  to  the 
third  shall  also  be  the  same  as  that  of  the  first  to  the  second,  or 
of  the  third  to  the  fourth,  we  shall  have  a  series  of  numbers, 
each  of  which,  divided  by  the  preceding  one,  will  give  the  same 
ratio.  Hence,  if  any  term  be  multiplied  by  this  quotient,  the 
product  will  be  the  succeeding  term.  A  series  of  numbers  so 
formed  is  called  a  geojnetrical  progression.     Hence, 

A  geometrical  progression,  or  progression  by  quotients,  is  a  series 
of  terms,  each  of  which  is  equal  to  the  product  of  that  which 
precedes  it  by  a  constant  number,  which  number  is  called  the 
ratio  of  the  progression.     Thus,  in  the  two  series, 

3,     6,  12,  24,  48,  96,  .  .  . 

64,  16,     4,     1,  -1,  1,  .  .  . 
4     16 

each  term  of  the  first  contains  that  which  precedes  it  tvnce,  or 
is  equal  to  double  that  which  precedes  it ;  and  each  term  of  the 
second  contains  the  term  which  precedes  it  one-fourth  times,  oi 
is  a  fourth  of  that  which  precedes  it.     These  are  geometrical  pro- 


186  ELEMENTS    OF    ALGEBRA.  [CHAP.   VII. 

gressions.  In  the  first,  the  ratio  is  2 ;  in  the  second,  it  is  \. 
The  first  is  called  an  increasing  progression,  the  second  a  dc' 
creasing  progression. 

Let  a,  b,  c,  d,  e,  f,  .  .  .  denote  numbers  in  a  progression  by 
quotients  :  they  are  written  thus : 

a  :  b  :  c  :  d  :  e  :  f  :  g  .  .  . 

and  it  is  enunciated  in  the  same  manner  as  a  progression  by  dif- 
ferences. It  is  necessary,  however,  to  make  the  distinction,  thai 
one  is  a  series  of  equal  differences,  and  the  other  a  series  of 
equal  quotients  or  ratios.  It  should  be  remarked,  that  each  term 
of  the  progression  is  at  the  same  time  an  antecedent  and  a  con- 
sequent, except  the  first,  which  is  only  an  antecedent,  and  the 
last,  which  is  only  a  consequent. 

191.  Let  r  denote   the  ratio  of  the   progression 

a  :  b  :  c  :  d  .  .  .  ; 

r  being  >  1  when  the  progression  is  increasing,  and  r  <  1  when 
it  is  decreasing.  We  deduce  from  the  definition,  the  following 
equations  : 

b  =  ar,     c  =  br  =  ar"^,     d  ^=  cr  ^  ar'^,     e  =z  dr  ^  ar*  .  .  . ; 

and,  in  general,  any  term  n,  that  is,  one  which  has  n  —  1  terms 
before  it,  is   expressed  by    cr""!. 

Let  I  be   this   term ;    we   have  the   formula 
I  =  ar''-\ 

by  means  of  which  we  can  obtain  any  term  without  being  obliged 
to  find  all  the  terms  which  precede   it.     That  is, 

Any  term  of  a  geometrical  progression  is  equal  to  the  first  term 
multiplied  by  the  ratio  raised  to  a  power  whose  exponent  denotes 
the  number  of  preceding  terms. 

EXAMPLES. 

1.  Find  the  5th  term  of  the  progression 
2  :  4  :   8  :   16,  &c., 
in   which  the   first  terra  is   2,   and  the   common  ratio  2. 
5th  term    =2  x  2*  =  2  X  16  =  32. 


CilAP.   VII.]  GEOMETRICAL    PROGRESSION.  187 

2.  Find  the   8th  term  of  the  progression 

2  :  6  :   18  :  54  .  .  . 
8th  term    =  2  X  3^  =  2  X  2187  =  4374. 

3.  Find  the   12th  term  of  the  progression 

64  :   16  :  4  :   1   :  —  .  . 

4 

/  1  \^i        43  1  1 

12th  term    =  64  (  — )    =-—  =  —  =  — — . 
V4/  411        48       65536 

192.  We  will  now  explain  the  method  of  determining  tlie  sura 
of  n  terms  of  the  progression 

a  :  b  :  c  :  d  :  e  :  f  :  .  .  .  :  i  :  k  :  I, 

of  which  the  ratio  is  r. 

If  we  denote  the  sum  of  the    series   by  S,  and   the  7ith.   term 
by  /,  we  shall  have 

S  =  a  +  ar  +  ar"^  .   .   .   .    +  ar"~'^  +  a?""~'. 

If  we  multiply  both  members  by  r,  we  have 

Sr  =  ar  -{-  ar^  +  ar^   .   .   .   +  a?'""^  +  ar" ; 

and  by  subtracting  the   first  equation, 

ar^  —  a 
Sr  —  S  =  ar"^  —  o,    whence,     S  =  —  ; 

T  —  \ 

and  by  substituting   for  a;-",  its   value   Ir,  we  have 

Ir  —  a 


S  = 


r  —  1 


That  is,  to  obtain  the  sum  of  any  number  of  terms  of  a  pro- 
gression by  quotients,  multiply  the  last  term  by  the  ratio,  subtract 
the  first  term  from  this  product,  and  divide  the  remainder  by  the 
ratio  diminished  by  unity. 

EXAMPLES. 

1.  Find  the  sum  of  eight  terms  of  the  progression 
2  :   6  :   18  :  54  :   162  .  .  .  :  2  X  3^  =  4374 

S  =  i^''-  =  illH?-^  =  6560. 
r  -  1  2 


188  ELEMENTS    OF    ALGEBRA.  [CHAP.   VII. 

2.  Find  the   suin  of  five   terms   of  the   progression 

2  :  4  :   8  :   16  :   32  ;  .  .  .  . 

r  —  1  1 

3.  Find  the  sum  of  ten  terms  of  the  progression 

2  :  6  :   18  :  54  :   162  .  .  .  2  X  3^  =  39366. 

Ans.  59048. 

4.  What  debt  may  be  discharged  in  a  year,  or  twelve  months, 
by  paying  $1  the  first  month,  $2  the  second  month,  $4  the  third 
month,  and  so  on,  each  succeeding  payment  being  double  the  last ; 
and  what  will  be   the   last  payment  1 

Ans.  Debt,  $4095 ;    last  payment,  $2048, 

5.  A  gentleman  married  his  daughter  on  New-Year's  day,  and 
gave  her  husband  Is.  toward  her  portion,  and  was  to  double  it 
on  the  first  day  of  every  month  during  the  year :  what  was  her 
portion  ?  Ans.  £204  15^. 

6.  A  man  bought  10  bushels  of  wheat  on  the  condition  that  he 
should  pay  1  cent  for  the  first  bushel,  3  for  the  second,  9  for 
the  third,  and  so  on  to  the  last:  what  did  he  pay  for  the  last 
bushel  and   for  the   ten  bushels  1 

Ans.   Last  bushel,   $196,83  ;    total  cost,  $295,24. 
193.   When  the  progression  is  decreasing,  we  have    r  <^  I    and 
/  <  ff ;    the  above   formula  for  the   sura  is  then  written  under  the 
form 

1  —  r 
in  order  that   the   two  terms  of  the   fraction   may  be   positive. 
By  substituting  ar"~'^   for  I  in  the  expression  for  S,  it  becomes 

ar'^  —  a  r^        «  —  '^^" 

b  =  — — -,     or     S  = . 

r  —  1  1  —  r 

EXAMPLES. 

1.  Find  the   sum  of  the  first  five  terms  of  the  progression 
32  :   16  :   8  :  4   :  2. 

32  —  2  X  —       „, 

S  =  °-^  = ?.  =21  =  63. 

1  -  r  1  1 


CHAP.   VII.]  GEOMETRICAL    PROGRESSION.  189 

2.  Find  the  sura  of  the  first  twelve   terms   of  the  progression 
64:16:4:l:-:...:64y    ,     or     -3-^. 

1  1  1 

64  -  — -—  X  —      256 


^^a  —  lr  65536       4  _  65536  _  65535 

^  ~  l^ir>  =  3  ~  3       -    ^  +  196608' 

~A 
We  perceive   that  the   principal   difficulty   consists    in   obtaining 
the   numerical  value  of  the    last  term,  a   tedious   operation,  even 
when  the  number  of  terras  is  not  very  great. 

194.  Remark. — If,  in  the  formula 

^^        T-\       ' 

we  suppose  r  =  1,  it  becomes 

This  result  is  a  symbol  of  indetermination.  It  often  arises 
from  the  existence  of  a  coraraon  factor  (Art.  113),  which  becomes 
nothing  by  making  a  particular  hypothesis  on  the  quantities  which 
enter  the  equation.  If  this  common  factor  can  be  divided  out,  the 
expression  will  assume  a  determinate  form.  This,  in  fact,  is  the 
case  in  the  present  question ;  for,  the  expression  r"  —  I  is  divisi- 
ble by    r  —  1    (Art.  61),  and  gives  the  quotient 

r»-i  +  ?-""^  4-  ?-"-3  +   .  .  .   +  r  +  1 ; 
hence,  the   value  of  S  takes  the  form 

S  =  ar''-^  +  ar'^-'^  +  ar'^-'^  +   ...   -\-  ar  ■\-  a. 
Now,  making   r  =  1 ,  we  have 

S  =  a  +  o  +  a+   •   •  •   +o  =  na. 
We  can  obtain  the  same  result  by  going  back  to  the  proposed 
progression 

a  :   6  :  c  :   .  .  .  :  /, 
which,  in  the  particular  case  of  r  =  1,  reduces  to 

a  :   a  :  a  \   .  .  .  '.  a, 
the  sum   of  which   series  is   equal   to  na. 


J  90  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIT 

The  result    — ,    given  by  the  formula,  may  be  regarded   as  in 

dicatinijf  that  the  series  is  characterized  by  some  particular  prop 
erty.  In  fact,  the  progression,  being  entirely  composed  of  equal 
terms,  is  no  more  a  progression  by  quotients  than  it  is  a  pro- 
gression by  differences.  Therefore,  in  seeking  for  the  sum  of  a 
certain  number  of  the  terms,  there  is  no  reason  for  using  the 
ibrmula 

a{r-  -I) 

^-      r-l     ' 
in  preference  to  the   formula 

{a  +  I)n 

^  -  2 • 

which  gives   the  sum  in   the   progression   by  differences. 

195.  The  consideration  of  the  five  quantities,  a,  r,  n,  I,  and  S, 
which  enter  into  the  formulas 

Ir  —  a 


I  =  ar"-i     and     S 


r-l' 
give  rise   to   several   curious   problems. 

Of  these  cases,  we  shall  consider  here,  only  the  most  im- 
portant. We  will  first  find  the  values  of  S  and  r  in  terms  of 
a,   Z,   and  n. 

The  first  formula  gives 

r"~i  =  — ,      whence       r  =  \/  — . 
a  »     a 

Substituting  this  value  in  the  second  formula,  the  value  of  S  will 
be  obtained. 

The  expression 


furnishes  the   means   for  resolving  the   following  question,  viz. : 
To  jiml  m   mean  proportionals  hettoeen  two  given  numbers  a  and 

b ;    that  is,  to  find  a   number  m  of  means,   which  will  form  with  a 

and  b,  considered  as   extremes,  a  progression  by  quotients. 

To    find   this    series,    it   is    only  necessary   to   know   the    ratio. 

Now,   the    required   number   of  means  being  m,   the  total  number 


CHAP.   VII.]  GEOMETRICAL    PROGRESSION.  19] 

of  terms  consiJereJ,  will  be  equal  to    m  +  2.     Moreover,  \vc  have 
I  =  b,   therefore  the   value   of  r  becomes 

*     a 

tliat  is,  we  must  divide  one  of  the  given  numbers  (b)  hy  the  other 
(a),  then   extract  that   root  of  the   quoticiit  whose  index   is  one  more 
than   the  required  number   of  means. 
Hence,  the  progression  is 

/    0  /    0  .     /    L* 

a  :  a  \/  —   :  a  \/  — ;^  :  a  \/  — -  :   .  .  .  0. 
*      a  ''a-  *     a-* 

Thus,  to   insert    six    mean    proportionals    between   the    numbers 
3   and  384,   we  make  to  =  G,   M'hence 


3S4        7  / 

=  V128  =  2; 


3 
whence  we  deduce  the   progression 

3  :  6  :   12  :  24  :  48  :   96  :   192  :  384. 

Remark. — When  the  same  number  of  mean  proportionals  are 
inserted  between  all  the  terms  of  a  progression  by  quotients,  taken 
two  and  two,  all  the  progressions  thus  formed  will  constitute  a 
single  progression. 

Of  Progressions  having  an  ivjlnite  Number  of  Terms. 
196.  Let  there  be  the   decreasing  progression 
a  :  b  :  c  :  d  :  e  :  f  '.  .  .  .y 
containing  an  indefinite  number  of  terms.     The  formula 

a  —  ar^ 
1  —  r 
which  represents  the  sum  of  n  terms,  can  be  put  under  the  form 

a  ar" 


S  = 


1  —  r        1  — r 

Now,  since  the  progression  is    decreasing,  r  is  a    proper    frac- 

ii(m,  and  r"  is  also   a  fraction,  which  diminishes  as   n  increases. 

Therefore,  the   greater  the    number  of  terms   we  take,   the    more 

uill X  r"    diminish,  and   consequently,  the    more  will    the 

]  —  r 


192  ELEMENTS    OF    ALGEBRA.  [CHAP.  VII 

partial   sum  of  these   terms   approximate   to    an   equality  with   the 

first    part    of   S  ;    that   is,    to    • .     Finally,  when   n   is    takec 

1  —  r 

greater  than   any  given  number,  or 

n  =  00,    tlien      X  r* 

1  —  r 

will  be  less  than  any  given  number,  or  will  become  equal  to  0  , 

and  the  expression    will   represent  the   true   value   of  the 

1  —  r 

sjim  of  all  the  terms   of  the  series. 

Whence,  we  may  conclude,  that  the   expression  for  the  sum  of 

the  terms  of  a  decreasing  progression,  in  which  the  number  of  terms 

is  infinite,  is 

1  —r 
This  is,  properly  speaking,  the  limit  to  which  the  partial  sums 
approach,  by  taking  a  greater  number  of  terms  of  the  progression. 
The  number  of  terms  may  be   taken  so  great  as  to  make  the  dif- 

ference  between  the  sum,  and  -^-,    as  small  as  we  please,  and 

1  —  r 

the  difl'erence  will  only  become  nothing  when  the  number  of  terms' 

taken  is  infinite. 


EXAMPLES. 


1,  Find   the   sum  of 


1111  .  ^  . 

1   :  —  :  —  :  —  :  —    to  infinity, 
3         9       27      81  ^ 

We  have,  for  the  sum  of  the  terms, 

S  =  —-  =       ^  ^ 


1  -r       ,  _  J_       2 
3 

2.  Again,  take  the  progression 

1   :  —  :  —  :  —  :  —  :  —  :  &c.  .  .  , 

2        4        8       16      32 

We  have  S  =  — ^—  =  — ^—-  -  2. 

What  is  the  error,  in  each  example,  for   »  =  4,  /i  ==  5,  «  —  6  ■ 


C!ii4P.   VIIJ.l  PERMUTA.TIONS    AND    COMBINATIONS.  1  i3 


CHAPTER  VIII. 

FORMATION    OF  POWERS,  AND  EXTRACTION  OF  ROOTS  OF  ANY  DEGREE. 
CALCULUS    OF    RADICALS. INDETERMINATE    CO-EFFICIENTS 

197.  The  resolution  of  equations  of  the  second  de^ee  supposes 
the  process  for  extracting  the  square  root  to  be  known.  In  like 
manner,  the  resolution  of  equations  of  the  third,  fourth,  &c.  de- 
gree, requires  that  Ave  should  know  how  to  extract  the  third, 
fourth,  &c.  root  of  any  numerical   or  algebraic  quantity. 

The  power  of  a  number  can  be  obtained  by  the  rules  of  mul- 
tiplication, and  this  power  is  subjected  to  a  certain  law  of  forma- 
tion, which  it  is  necessary  to  know,  in  order  to  deduce  the  root 
from  the  power. 

Now,  the  law  of  formation  of  the  square  of  a  numerical  or 
algebraic  quantity,  is  deduced  from  the  expression  for  the  square 
of  a  binomial  (Art.  116);  so  likewise,  the  law  of  a  power  of  any 
degree,  is  deduced  from  the  same  power  of  a  binomial.  We  shall 
therefore  first  determine  the  development  of  any  power  of  a  binomial. 

198.  By  multiplying  the  binomial  x-i-a  into  itself  several  times, 
the  following  results  are  obtained : 

(x  -\-  a)    =z  X   -\-  a, 

(x  4-  a)2  =  x'^  +  2ax    +  a^, 

(x  +  a)^  =:  x^  -\-  3ax^  -f-  3a~x  +  a^i 

(x  +  a)*  =  a;*  4-  4ax3  -|-  da'^x'^  +  4a^x  +  a*, 

(x  +  a)5  —  x^  -}-  5ax^  +  lOa-x^  +  lOa^x^  -f  5a* x  -f  a\ 

By   examining   the    developments,  Ave    readily  discover    the   law 

according  to  which  the  exponents  of  x  decrease   and  those   of  a 

increase,  in  the  successiA^e  terms  ;   it  is  not,  howcA-er,  so  easy  to 

13 


JV4  ELEMENTS    OK    ALGEBRA.  [CHAP.   V1.U. 

discover  a  law  for  the  co-efficients.  Newton  discovered  one,  by 
means  of  which  a  binomial  may  be  raised  to  any  power,  without 
first  obtaining  all  of  the  inferior  powers.  He  did  not,  however, 
explain  the  course  of  reasoning  which  led  hin\  to  the  disc  )very ; 
but  the  law  has  since  been  demonstrated  in  a  rigorous  manner. 
Of  all  the  known  demonstrations  of  it,  the  most  elementary  is 
that  which  is  founded  upon  the  theory  of  combinations.  However, 
as  the  demonstration  is  rather  complicated,  we  will,  in  order  to 
simplify  it,  begin  by  resolving  some  problems  relative  to  permuta- 
tions and  combinations,  on  which  the  demonstration  of  the  formula 
for  the  binomial  theorem  depends. 

Theory  of  Pcrmutatiojis  and   Combinations. 

199.  Let  it  be  proposed  to  determine  the  whole  number  of  loays 
in  which  several  letters,  a,  b,  c,  d,  &c.,  can  be  written  one  after 
the  other.  The  result  corresponding  to  each  change  in  the  posi- 
tion of  any  one   of  these  letters,  is   called  a  permutation. 

Thus,  the  two  letters  a  and  b  furnish  the  two  permutations,  ab 
and  ba. 

abc 
acb 
cab 
bac 
bca 
.cba 

Permutations,  are  the  results  obtained  by  writing  a  certain  num- 
ber of  letters  one  after  the  other,  in  every  possible  order,  in  such  a 
manner  that  all  the  letters  shall  enter  into  each  result,  and  each 
letter  enter  but  once. 

Problem  1.  To  determine  the  number  of  permutations  of  which 
n  letters  are  susceptible. 

In  the    first   place,  two  letters,  a  and  b,  evidently  (  ab 

give  two  permutations.  (   ba 

Therefore,  the  number  of  permutations  of  two  letters  is  ex 
pressed  by  1  x  2. 

Take    the    three    letters,    a,    b,    and    c.      Reserve  r  c 

either  of  the    letters,   as    c,  and    permute    the    other  }    ab 

two,  giving  '    ba 


In  like  manner,  the  three  letters,   a,  b,  c,  furnish 
six  permutations. 


CHAP.   VIII.]  PERiMOTATIONS    AXD    COMBINATIONS.  195 


cab 
acb 
ahc 
chn 
hca 
_bac 


Now,  the  third  letter  c  may  be  placed  before  ah, 
between  a  and  h,  and  at  the  right  of  ab  ;  and  the 
same  for  ba  :  that  is,  in  one  of  the  first  permuta- 
tions, the  reserved  letter  c  may  have  three  different 
■places,  giving  three  permutations.  Now,  as  the  same 
may  be  shown  for  each  one  of  the  first  permutations, 
it  follows  that  the  whole  number  of  permutations  of 
three  letters  will  be  expressed  by,   1  X  2  X  3. 

If  now,  a  fourth  letter  d  be  introduced,  it  can  have  four  places 
in  each  one  of  the  six  permutations  of  three  letters  :  hence,  all 
the  permutations  of  four  letters  ^vill  be  expressed  by,  1  x  2  x  3  x  4. 

In  general,  let  there  be  n  letters,  a,  b,  c,  &c.,  and  suppose  the 
total  number  of  permutations  of  n  —  1  letters  to  be  known  ;  and 
let  Q  denote  that  number.  Now,  in  each  one  of  the  Q  permuta- 
tions, the  reserved  letter  may  have  n  places,  giving  n  permutations  : 
hence,  when  it  is  so  combined  with  all  of  them,  the  entire  num- 
ber of  permutations   will   be   expressed   by   Q  x  n. 

Let  n^2.  Q  will  then  denote  the  number  of  permutations  that 
can  be  made  with  a  single  letter;  hence,  Q  =  1,  and  in  this 
particular  case   we   have,    Q  x  n  =  1  X  2. 

Let  71  =  3.  Q  will  then  express  the  number  of  permutations 
of  3  —  1  or  2  letters,  and  is  equal  to  1x2.  Therefore,  Q  X  n 
is  equal  to  1  x  2  x  3. 

Let  n  =  4.  Q  in  this  case  denotes  the  number  of  permutations 
of  3  letters,  and  is  equal  to  1  x  2  x  3.  Hence,  Q  x  n  becomes 
1  X  2  X  3  X  4 ;  and  similarly,  when  there   are   more  letters. 

200.  Suppose  we  have  a  number  m,  of  letters  a,  b,  c,  d,  &c. 
If  they  are  written  one  after  the  other,  in  classes  of  2  and  2,  or 
3  and  3,  or  4  and  4  ...  in  every  possible  order  in  each  class, 
in  such  a  manner,  however,  tliat  the  number  of  letters  in  each 
result  shall  be  less  than  the  number  of  given  letters,  we  may  de- 
mand the  whole  number  of  results  thus  obtained.  These  results 
are   called   arrangements.  > 

Thus,  ab,  ac,  ad,  .  .  .  ba,  be,  bd,  .  .  .  ca,  cb,  cd,  .  .  .  are  ar- 
rangements of  m  letters   taken  2  and  2  ;    or  in  sets   of  2   each. 

In  like  manner,  abc,  abd,  .  .  .  bac,  bad,  .  .  .  acb,  acd,  .  .  .  are 
arrangements   taken  in   sets  of  3. 

Arrangements,  are  the  results  obtained  by  writing  a  number  m  of 
Utters  one  after  the  other  in  every  possible  order,  in   sets  of  2  and 


196  ELEMENTS    OF    ALGEBRA.  [CHAP.   VIII. 

2,  3  and  3,  4  and  4  .  .  .  n  and  n  ;  m  being  >  n  ;  lliat  is,  the 
number  of  letters  in  each  set  being  less  than  the  whole  number 
of  letters  considered.  If,  however,  we  suppose  n  =  m,  the  ar- 
rangements taken  n  and  n,  will  become  simple  permutations . 

Problem  2.  Having  given  a  number  m  of  letters  a,  b,  c,  d  .  .  . 
to  determine  the  total  number  of  arrangements  that  may  he  formed  of 
them  by  taking  them  n  at  a  time ;  m  being  supposed  greater  than  n. 

Let  it  be  proposed,  in  the  first  place,  to  arrange  the  three  let- 
ters, a,  b,  and  c,  in  sets  of  two  each. 

First,  arrange  the  letters  in  sets  of  one  each,  and  for 
each  set  so  formed,  there  will  be  two  letters  reserved  : 
the  reserved  letters  for  either  arrangement,  being  those 
which  do  not  enter. 

When  we  arrange  with  reference  to  a,  the  reserved  letters  will 
be  b  and  c ;  if  with  reference  to  b,  the  reserved  letters  will  be 
a  and  c,  &c. 

Now,  to  any  one  of  the  letters,  as  a,  annex,  in  suc- 
cession, the  reserved  letters  b  and  c :  to  the  second 
arrangement  h,  annex  the  reserved  letters  a  and  c ;  and 
to  the  third  arrangement,  c,  annex  the  reserved  letters 
a  and  b :  this  orives  ,     , 

CO 

And  since  each  of  the  first  arrangements  is  repeated  as  many 
times  as  there  are  reserved  letters,  it  follows,  that  the  arrange- 
ments of  three  letters  taken  two  in  a  set,  will  be  equal  to  the  ar- 
rangements of  the  same  number  of  letters  taken  one  in  a  set,  multi- 
plied by  the  number  of  reserved  letters. 

het  it  be  required  to  form  the  arrangement  of  four  letters, 
a,  b,  c,  and  d,  taken  3  in  a  set. 

First,  arrange  the  four  letters  in  sets  of  two :    there  fab 

will  then  be  two  reserved  letters.  Take  one  of  the 
sets  and  write  after  it,  in  succession,  each  of  the  re- 
served letters  :  we  shall  thus  form  as  many  sets  of 
three  letters  each  as  there  are  reserved  letters ;  and  these 
sets  differ  from  each  other  by  at  least  the  last  letter.  °  -^ 
Take  another  of  the  first  arrangements,  and  annex  in 
succession  the  reserved  letters  ;  we  shall  again  form 
as  many  different  arrangements  as  there  are  reserved 
letters.  Do  the  same  for  all  of  the  first  arrangements, 
and  it  is  plain,  that  the  whole  number  of  arrangements  L  (/  c 


ab 
ac 

ha 
he 
ca 


ba 
ac 
c  a 
ad 
da 
be 
ch 
hd 
dh 


CHAP.   VIII.]  PERMUTATIONS    AND    COMBINATIONS.  197 

which  will  be  formed,  of  four  letters,  taken  3  and  3,  will  he  equal 
to  the  arrangements  of  the  same  letters,  taken  two  in  a  set,  mul- 
tiplied hy  the  number  of  reserved   letters. 

Ill  order  to  resolve  this  question  in  a  general  manner,  suppose 
the  total  number  of  arrangements  of  m  letters,  taken  n  —  1  in  a 
set,   to   be   known,  and   denote   this   number   by  P. 

Take  any  one  of  these  arrangements,  and  annex  to  it,  in  suc- 
cession, each  of  the  reserved  letters,  and  of  which  the  number 
is  m  —  [ii  —  1),  or  m  —  ra  +  1  :  it  is  evident,  that  we  shall  thus 
form  a  number  m  —  ra  +  1  of  ^^w  arrangements  of  n  letters,  each 
differing  from  the  other  by  the  last  letter.  Now,  take  another  of  the 
first  arrangements  of  w  —  1  letters,  and  annex  to  it,  in  succession, 
each  of  the  m  —  n  -\-  1  letters  which  do  not  make  a  part  of  it ;  we 
again  obtain  a  number  rn  —  »  +  1  of  arrangements  of  n  letters, 
differing  from  each  other,  and  from  those  obtained  as  above,  at  least 
in  one  of  the  n  —  \  first  letters.  Now,  as  we  may  in  the  same 
manner,  take  all  the  P  arrangements  of  the  m  letters,  taken  n  —  1 
in  a  set,  and  annex  to  each  in  succession  each  of  the  m  —  n  +  1 
other  letters,  it  follows  that  the  total  number  of  arrangements  of 
m  letters  taken  n  in  a  set,  is  expressed  by 

P  {m  —  n  -I-  1). 

To  apply  this  in  determining  the  number  of  arrangements  of 
m  letters,  taken  2  and  2,  3  and  3,  4  and  4,  or  5  and  5  in  a  set, 
make  ra  =  2  ;  whence,  m  —  n  -\-  \  ^^  m  —  1;  Pin  this  case, 
will  express  the  total  number  of  arrangements,  taken  2  —  1  and 
2  —  1,  or  1  and  1  ;  and  is  consequently  equal  to  '  m  ;  therefore, 
the   formula  becomes    m{in  —  1). 

Let  n  =  3  ;  whence,  m  —  n+l=??j  —  2;  P  will  then  ex- 
press the  number  of  arrangements  taken  2  and  2,  and  is  equal  to 
m[m—V)\   therefore,  the   formula  becomes 

m{in  —  1)  (rn  —  2). 

Again,  take  n  =  4  :  whence,  m  —  w+l=m  —  3:  P  will  ex- 
press the  number  of  arrangements  taken  3  and  3,  or  is   equal  to 

m,{m  —  1)  (^  —  2) ; 

:.Lerefore,  the  formula  becomes 

»i  (m  —  1)  (»i  —  2)  (m  —  3). 


198  ELEMENTS    OF    ALGEBRA.  [CHAP.   VIII. 

Remark. — From  the  manner  in  which  these  results  have  been 
deduced,  we  conclude  that  the  general  formula  for  m  letters  taken 
n   in   a  set,   is 

m  (ot  —  1)  (m  —  2)  (m  —  3)   .   .   .   .   (m  —  n  +  1)  ; 

that   is,  it  is   composed  of  the  product  of  the   n  consecutive  numbers 
comprised  between  m  and   m  —  n  +  1,    inclusively. 

From  this  formula,  that  of  the  preceding  Art.  can  easily  be 
deduced,  viz.,  the  development  of  the  value  of  Q  x  n. 

For,  we  see  that  the  arrangements  become  permutations  when 
the  number  of  letters  entering  into  each  arrangement  is  equal  to 
the  total  number  of  letters  considered. 

Therefore,  to  pass  from  the  total  number  of  arrangements  of  m 
letters,  taken  n  and  n,  to  the  number  of  permutations  of  n  letters, 
it  is  only  necessary  to  make  w  =  «  in  the  above  development, 
which  gives 

n(«  -  1)  («  -2)  (n-  3) 1. 

By  reversing  the  order  of  the  factors,  and  observing  that  the 
last  is  1,  the  next  to  the  last  2,  the  third  from  the  last  3,  &c., 
we  have 

1x2x3x4 {n-2){n-\)n, 

for  the  development  of   Q  x  ri. 

This  is  nothing  more  than  the  series  of  natural  numbers  com- 
prised between  1    and  n,  inclusively. 

201.  When  the  letters  are  disposed,  as  in  the  arrangements,  2 
and  2,  3  and  3,  4  and  4,  &c.,  it  may  be  required  that  no  two  of 
the  results,  thus  formed,  shall  be  composed  entirely  of  the  same 
letters,  in  which  case  the  products  of  the  letters  will  be  different  j 
and  we  may  then  demand  the  whole  number  of  results  thus  ob- 
tained.    In  this  case,  the  results   are   called  combinations. 

Thus,  ab,  ac,  he,  .  .  .  ad,  bd,  .  .  .  are  combinations  of  the  let- 
ters a,  b,  and  c,  Sic,  taken  2  and  2. 

In  like  manner,  abc,  abd,  .  .  .  acd,  bed,  .  .  .  are  combinations 
of  the  letters  taken  3  and  3. 

Combinations,  are  arrangements  in  which  any  two  will  differ  from 
each  other  by  at  least  one  of  the  letters  which  enter  them. 

Hence,  there  is  an  essential  difference  in  the  signification  ol 
the  words,  permutations,  arrangements,  and  combinations. 


CHAP.   VIII.]  PLRMUTATIOXS    AND    COMBINATIONS.  199 

Problem  3.  To  determine  the  total  number  of  different  combina- 
tions that  can  be  formed  of  m  letters,  taken   n   in  a   set. 

Let  X  denote  the  total  number  of  arrangements  that  can  be 
formed  of  m  letters,  taken  n  and  n  :  Y  the  number  of  permuta- 
tions of  n  letters,  and  Z  the  total  number  of  different  cornbinations 
taken  n  and  7i. 

It  is  evident,  that  all  the  possible  arrangements  of  m  letter?, 
taken  ra  in  a  set,  can  be  obtained,  by  subjecting  the  n  letters  of 
each  of  the  Z  combinations,  to  all  the  permutations  of  which 
these  letters  are  susceptible.  Now,  a  single  combination  of  n  let- 
ters gives,  by  hypothesis,  Y  permutations  ;  therefore  Z  combina- 
tions will  give  Y  X  Z  arrangements,  taken  ,i  and  n  ;  and  as  X 
denotes  the  total  number  of  arrangements,  it  follows  that  the  three 
quantities,  X,  Y,  and  Z,  give  the  relations 

X  =  Y  X  Z ;    whence,    Z  =  -:r=. 

But  we  have  (Art.  200), 

X=  P(m-n+  1), 

and  (Art.  199),  Y  =  Q  x  n  ; 

^       P(m  —  n+l)        P       m  —  n+l 

therefore,        Z  =  — ^— ^^— -'  =  -77  X  ^!—. 

Q  X  n  Q  n 

Since  P  expresses  the  total  number  of  arrangements,  taken 
n  —  1    and   n  —  1 ,    and    Q  the  number  of  permutations   of  n  —  1 

P 

letters,  it  follows  that    —    expresses  the  number  of  different  cora- 

binations  of  m  letters  taken  n  —  1  and  n  —  1. 

To    apply  this    to    the    case    of  the    combinations    of  m   letters 

taken  2  and  2,  3  and  3,  4  and  4,  &c. 

P 
Make  n  =  2,  in  which  case,  —   expresses  the  number  of  com- 

binations  of  m  letters  taken  2  —  1  and  2—1,  or  taken  1  and  1, 
and  this  number  must  be  equal  to  m ;  the  above  formula  there- 
fore becomes 

m  —  1  m  (m  —  1) 

m  X  or     —  ^ -. 

2  1.2 

P       . 

Let   n  =:  3  ;    —   will  express  the  number  of  combinations  taken 


200  ELEMENTS    OF    ALGEBRA.  'CHAP.  VIII. 

mhn  —  1 ) 
2  and  2,  and  is    equal  to     -— ;     and  the   forunua   becomes 

t  m[m  —  1)  (w  —  2) 

r2T3  * 

In  like  manner,  we  find  the  number  of  combinations  of  m.  let- 
ters taken  4  and  4,  to  be 

m{m  —  \)  (ot  —  2)  (ct  —  3)  ^ 
L2^3^4  ' 

and,  in  general,  the  number  of  combinations  of  m  letters  taivcn  n 
and  n,  is  expressed  by 

m  (m  —  1)  (wz  —  2)  (m  —  3)   .   .   .  {m  —  n  -\-  1) 
L2.3.4   .   .   .  {;j—  l).n  ' 

which  is  the  development  of  the  expression 

P(?n  — n4-  1) 
Q  Xn 

We  may  here  observe  that,  if  we  have  a  series  of  numbers, 
decreasing  by  unity,  and  of  which  the  first  is  m  and  the  last 
m — p,  m  and  j5  being  entire  numbers,  that  the  product  of  these 
numbers  will  be  exactly  divisible  by  the  continued  product  of  all 
the  natural  numbers   from   1    to  p  +  1    inclusively ;    that  is, 

w?  (?M  —  I)  (m  —  2)  (m  —  3)   .   .   .  {m  —  p) 

1      .     2       '.       3        '.       4      :  .   .  .  {JTl) 

is  a  whole  number.  For,  from  what  has  been  proved,  this  ex- 
pression represents  the  number  of  different  combinations  that  can 
be  formed  of  m  letters  taken  in  sets  of  />  +  1  and  p  -{-  I.  Now 
this  number  of  combinations  is,  from  its  nature,  an  entire  number ; 
therefore  the  above  expression  is  necessarily  a  whole  number. 

Demonstration  of  the   Binomial   Theorem. 

202.  In  order  to  discover  more  easily  the  law  for  the  develop- 
ment of  the  with  power  of  the  binomial  sc  -\-  a,  let  us  observe 
the  law  of  the  product  of  several  binomial  factors,  x  -\-  a,  x  -^  b, 
X  +  c,  X  -\-  d  .  .  .  of  which  the  first  term  is  the  same  in  each, 
and  the  second  terms  different. 


CHAP.  VIII.] 

BINOiMTAl 

THEOREM. 

X    -}-   a 

X   +  b 

X 

+ 

1st  product 

-     a;2  4-   a 

ab 

+   b 

X    -}-   c 

2d     -       - 

-     x^  -{-   a 

'x^ 

+ 

lib 

X    -f   abc 

+   b 

+ 

ac 

+   c 

+ 

be 

X   -\-  d 

3d     -       - 

'     X*  +   a 

,t3 

+ 

ab 

a^  +   abc 

+   b 

+ 

ac 

+  abd 

+   c 

+ 

ad 

-\-    acd 

+  d 

4- 
+ 
+ 

be 
bd 
cd 

+    bed 

201 


X   +   abed 


From  these  products,  obtained  by  the  common  rule  for  alge- 
braic multiplication,  we  discov^er  the  following  laws  : — 

1st.  With  respect  to  the  exponents,  we  observe  that,  the  ex- 
ponent of  X,  in  the  first  term,  is  equal  to  the  number  of  binomial 
factors  employed.  In  each  of  the  following  terms  to  the  right, 
this  exponent  diminishes  by  unity  to  the  last  term,  where  it  is  0. 

2d.  With  respect  to  the  co-efficients  of  the  different  powers  of 
a,',  that  of  the  first  term  is  unity  ;  the  co-efficient  of  the  second 
term  is  equal  to  the  sum  of  the  second  terms  of  the  binomials  ; 
the  co-efficient  of  the  third  term  is  equal  to  the  sum  of  the  prod- 
ucts of  the  different  second  terms,  taken  two  and  two ,  the  co- 
efficient of  the  fourth  term  is  equal  to  the  sum  of  their  different 
products,  taken  three  and  tliree.  Reasoning  from  analogy,  we  may 
conclude  that  the  co-efficient  of  the  term  which  has  n  terms  be- 
fore it,  is  equal  to  the  sum  of  the  different  products  of  the  second 
terms  of  the  m  binomials,  taken  n  and  n.  The  last  term  of  the 
product  is  equal  to  the  continued  product  of  the  second  terms  of 
the  binomials. 

In  order  to  prove  that  this  law  of  formation  is  general,  suppose 
that  it  has  been  proved  true  for  a  number  m  of  binomials  ;  let  us 


202  ELEMENTS    Of    ALGEBRA.  [CHAP.   VIII. 

see  if  it  will  continue  to  be  true  when  the  product  is  multiplied 
by  a  new  factor. 

For  this  purpose,  suppose 

to  be  the  product  of  ?«  binomial  factors,  AV"~"  representing  the 
term  which  has  n  terms  before  it,  and  ilfT'"~"+i  the  term  which 
immediately  precedes. 

Let    a:  +  ^    be  the  new  factor  by  which  we  multiply  ;  the  prod- 
uct when  arranged  according  to  the  powers  of  x,  will  be 


+  B    I  a;"'-^  +  C 
-{-  Ak\  +  Bk 


+  .  .  .  +N 
-\-Mk 


-hUk. 


From  which  we  perceive  that  the  law  of  the  exi-)onents  is  evi 
dently  the  same. 

With  respect  to   the  co-efficients,  we  observe, 
1st.  That  the  co-efficient  of  the  first   term  is  unity ;    and 
2d.  That    A  -\-  k,  or   the  co-efficient  of  a"',   is    the    sum  of  the 
second  terms  of  the   m  -j-  1    hiiiomials. 

3d.  Since,  by  hypothesis,  B  is  the  sum  of  the  different  products 
of  the  second  terms  of  the  m  binomials,  taken  two  and  two,  and 
since  A  x  k  expresses  the  sum  of  the  products  of  each  of  the 
second  terms  of  the  m  binomials  by  the  new  second  term  k ;  there- 
fore, B  -f  Ak  is  the  sum  of  the  different  products  of  the  second, 
terms  of  the    m  +  1    binomials,  taken  two  and  two. 

In  general,  since  N  expresses  the  sum  of  the  products  of  the 
second  terms  of  the  m  binomials,  taken  n  and  n,  and  M  the  sum 
of  their  products,  taken  n  —  1  and  n  —  1  ;  if  we  multiply  the 
last  set  by  the  new  second  term  k,  then  N  -J-  Mk,  or  the  co-effi- 
cient of  the  term  which  has  n  terms  before  it,  will  be  equal  to 
the  sum  of  the  different  products  of  the  second  terms  of  the 
m  -\-  I  binomials,  taken  n  and  n.  The  last  term  is  equal  to  the 
continued  product  of  the  second  terms  of  the   m  +  1    binomials. 

Therefore,  the  law  of  composition,   supposed  true  for  a  number 
m  of  binomial  factors,  is  also  true  for  a  number  denoted  hy  m  -{-  I 
Hence,  it  is  true  for  ?n  +  2,  &c.,  and  is  therefore  general. 

203.  Let  us  now  suppose,  that  in  the  product  resulting  fron 
he   multiplication  of  the   m  binomial  factors, 

X  -\-  a,    X  -\-  b,    X  -\-  c,    X  -}•  d,  .  .  .  . 


CHAP.    VIII.]  BINOMIAL    THEOREM.  203 

we  make,  a  =:  fj  —  c  ^^  d:   .   .   .  . 

we  shall  then  have 

(x  +  a)  (x  +  6)  (,T  +  c) =(x  +  a)'". 

The  co-efficient  of  the  first  term,  x"",  will  become  1.  The  co-effi- 
cient of  a'"'"^  being  a  +  i  +  c  +  J,  .  .  .  will  be  a  taken  7?i  times  ; 
that  is,   ma.     The   co-efficient  of  cif~^,  being 

ab  -\-  ac  -\-  ad  .   .   .   .  reduces  to    a^  _j_  ^2  -|-  a^  .   .   . 
that    is,   it  becomes    a^    taken   as   many  times   as   there   are   com- 
binations   of  m   letters,    taken    two    and   two,    and   hence    reduces 
(Art.  201),  to 

m  —  I     2 


2 
The   co-efficient  of  a;'""^  reduces   to  the  product  of  a^,  multiplied 
by  the  number  of  different  combinations  of  m  letters,  taken  three 
and  three  ;    that  is,  to 

m  —  1       m  —  5 


&c. 


2  3 

In    general,  let   us  denote    the  term,  which  has  n  terms  before 

it,  by  Nx"'~'^.     Then,  the   co-efficient   A''  will   denote  the   sum   of 

the   products  of  the   second  terms,  taken  n  and  n ;  and  when  all 

of  the  terms  are  supposed  equal,  it  becomes  equal  to  a"  multiplied 

by  the  number  of  different  combinations  that  can  be  made  with  m 

letters,  taken  n  and  n.     Therefore,  the  co-efficient   of  the   general 

term  (Art.  201),  is  p (m  -  n  +  I) 

iV.=  — ^—- a", 

Q  X  71 

from  which  we  deduce  the  formula, 

m  —  1 
{x  +  a)'"  =  x'"  -f  max'"~^  +  ?n  .  — _ w^ar'""^ 

771  — 1     m— 2    ,       ,  P(7n  —  7i  +  l) 

4-  m  . . «3^'"-3  .  .  .  -4 ^~-- a»j;'"-"  .  .  .  +  a"". 

2  3  Q  .  n 

The  term 

P  (m  —  n  +  \) 

5^ ;^ L  Qi^m-n 

Qn 
IS  called  the   general    ter7n,  because    by  making    n  =  2,  3,  4, 
all  of   the  others   can  be  deduced  from  it.     The  term    which  im- 
mediately precedes  it,  is  evidently. 


201  ELEMENTS    OF    ALGEBRA.  [CHAP.    VIIl. 

expresses  the    number  of  combinations  of  m  letters   taken    n  —  1 
and   n  —  1.     Hence,  we  see,   that  the   co-efficient 

Q  X  n 

P 

n   equal   to   the  co-efficient    —    of  the  preceding  term,  muhipUed 

by   m  —  n  -\-  1,  the  exponent  of  x  in  that  term,   and    divided   by 
«,  the  number  of  terms  preceding  the  required  term. 

P     . 

Since    —    is  the  co-efficient  of  the  preceding  term,  we  may,  by 

observing  how  the   co-efficients   are   formed    from  each   other,  ex- 
press the  co-efficient  of  the  general  term  thus, 

_  m  {in  —  ])  (in  —  2)  (m  —  3)   .   .   .  {m  —  n -{- 2)  (m  —  n  +  I) 

^    ~  1      r~2       '.       3        '.        4         T.         {n  —  f)      ;  n  '' 

The  simple  law,  demonstrated  above,  enables  us  to  determine 
the  co-efficient  of  any  term  from  the  co-efficient  of  the  preceding 
term. 

The  co-rffi,cicnt  of  any  term  is  formed  by  multiplying  the  co-cfp,- 
cient  of  the  preceding  term  by  the  exponent  of  x  in  that  term,  and 
dividing  the  product  by  the  number  of  terms  which  precede  the  re- 
quired term. 

For  example,  let  it  be  required  to  develop   (a;  +  of- 

From  this  law,  we  have, 

(x  -f  ay  =  a:«  +  Qax^  +  Iba"^ x^  +  20d^x^  +  15a%2  _[_  q^^^  -\-  a^. 

After  having  form.ed  the  first  two  terms  from  the  general 
formula  x^ -\- max'^'~'^ -{- ,  .  .  .  multiply  6,  the  co-efficient  of  the 
second  term,  by  5,  the  exponent  of  x  in  that  term,  and  then 
divide  the  product  by  2,  which  gives  15  for  the  co-efficient  of 
the  third  term.  To  obtain  that  of  the  fourth,  multiply  15  by  4, 
the  exponent  of  x  in  the  third  term,  and  divide  the  product  by  3, 
the  number  of  terms  which  precede  the  fourth  ;  this  gives  20  ; 
and  the  co-efficients  of  the  other  terms  are  found  in  the  same 
way. 

In  like  manner,  we  find 

(,^  _^  a)io  =  ^\Q  ^  iQax^  +  45a2a;8  j^  \20a^x''  +  2l0a^x^ 


CHAP.   VIII.]  BINOMIAL    THEORE.M.  203 

204.  It  frequently  occurs  that  the  terms  of  the  binomial  arc 
affected  with  co-efficients  and  exponents,  as  in  the  following  ex- 
ample : 

Let  it  be  required  to  raise  the  binomial 
3a"c  —  2bd 
to  the  fourth  power. 

Placing  Ba^c  =  x    and     —  2bd  =  y,     we  have 

(a;  +  yY  =  x*  -\-  4x^y  -j-  6x^)/^  -j-  Axy^  +  y*  ; 
and  substituting  for  x  and  y  their  values,  we  have 

(3a2c  -  2hdy  =  (Sa^c)*  +  4  (3a2c)3  (_  2bd)  +  6  (Sa'^cf  (-  2bd)^ 
+  4  (3a2c)  (_  2bdy  +  (-  2bdy, 
or,  by  performing  the  operations  indicated, 

(3a^c  —  2bdY  =  81a^c*  —  2l6a^c^d  +  216a*cH^d^  —  96a'^cb^d^ 
+  I6b*d\ 

The  terms  of  the  development  are  alternately  plus  and  minus, 
as  they  should  be,  since  the  second  term  is  — . 

205.  The  poAvers  of  any  polynomial,  may  easily  be  found  bv 
the  binomial  theorem.     For  example,  raise 

a  -\-  b  -{•  c 
to  the  third  power. 

First,  put  b  -\-  c  z=  d. 

Then       {a -{- b  +  cf  =  (a  +  df  =  a^  -\-  2aH  +  SarP  +  J3 . 

and  by  substituting  for  the  value  of  d, 

(a  +  6  +  c)3  =  a3  -f  2an  +  Sai^  _^  ^3 

3a2c  +  362c  +  Qahc 
+  3ac2  4-  35c2 
-f  c3. 
This  development  is   composed  of  the  cubes  of  the   three  ter??is, 
plus  three  times  the  square  of  each  term  by  the  first  powers  of  the 
two  others,  plus  six  times  the  product  of  all  three  terms. 

To  apply  the  preceding  formula  to  the  development  of  the  cui:e 
of  a  trinomial,  in  which  the  terms  are  affected  with  co-efficieiif.s 
and  exponents,  designate  each  term  by  a  single  letter,  and  perfurm 
the  operations  indicated;  then  replace  the  letters  introduced  by  their 
values 


20G  ELEMENTS    OF    ALGEBRA.  [CHAP.   VIII, 

From  this  rule,  we  will  find  that 

(2a2  _  4ab  +  3^2)3  ^  Sa<^  —  48a^b  +  \32a^P  -  208a^^ 
+  I98a-b^  —  ]08ab^  +  27b^. 
The  fourth,  fifth,  &c.  powers  of  any  polynomial  can  be  developed 
m  a  similar  manner. 

Conseqiiences  of  the  Binomial  Formnla. 

206.  The  development  of  the  binomial  expression  (x  -\-  a)'^ 
will  always  contain  ?w  +  1  terms.  Hence,  if  we  take  that  term 
of  the  development  which  has  ?i  terms  before  it,  the  number  of 
terms   after  it  will  be   expressed  by  m  —  n. 

Let  us  now  seek  the  co-efficient  of  the  term  which  has  n  terms 
after    it,    and    which,    consequently,  has    m  —  n    terms   before    it. 
We   obtain   this   co-efficient   by  simply  substituting   m  —  n   for    ra, 
in  the  last  value   of  N  in   Art.  203.     We  then   have, 
_  m{m-  l)(m-2)  .   .  (n  +  2)  (?z  +  1) 
~  1    .    2    .    3   .   .   .  (m  —  ?t  —  1)  (m  —  ?i)' 

As  we  can  always  take  the  term  which  has  n  terms  before  it, 
nearer  to  the  first  term  than  the  one  which  has  m  —  n  terms  be- 
fore it,  we  will  examine  that  part  of  the  co-efficient  which  is 
derived  from  the  terms  lying  between  these  two.     We  may  write 

m{m—\).  .  .  {m  —  n\-\).{m  —  n).{m-n-\).  .  .  (n  +  2) .  (n+1) 
1    .  2  n  («+l)  •  (w  +  2)  .  .  {in  —  n  —  \)\m-i{) 

Now,  by  cancelling  the  like  factors  in  the  numerator  and  de- 
nominator, we  have 

^^        m  {in  —  1)   ....   (/7j  —  n  4-  1) 

N  =  — ^^ :    hence, 

1.2     n 

In  the  development  of  any  power  of  a  binomial,  the  co-ejficients 
at  equal  distances  from  the  two  extremes  are  equal  to  each  other. 

207.  If  we  designate  by  K  the  co-efficient  of  the  term  which 
has  n  terms  before  it,  that  term  will  be  expressed  by  Za"a;"'-'' ; 
and  the  corresponding  term  counted  from  the  last  term  of  the 
series,  will  be    Ka'^~~"x'\ 

Now,  the  first  co-efficient  expresses  the  number  of  different  com- 
binations that  can  be  formed  with  m  letters  taken  n  and  n ;  and 
the  second,  the  number  which  can  be  formed  when  taken  m  —  n 


CHAP.   VIII.]  CUBE    ROOT    OF    NUMBERS.  207 

and  m  —  n  ;  we  may  therefore  conclude  that,  the  miniher  of  dif- 
ferent combinations  of  m  letters  taken  n  and  n,  is  equal  to  the  num- 
ber of  combinations  of  m  letters  taken  m  —  n  and  m  —  n. 

For  example,  twelve  letters  combined  5  and  5,  give  the  same 
number  of  combinations  as  when  taken  12  —  5  and  12  —  5,  or  7 
and  7.  Five  letters  combined  2  and  2,  give  the  same  number  of 
combinations   as   when   combined   5  —  2    and   5  —  2,   or   3   and   3 

208.  If,  in  the  general  formula, 

(x  +  a)'"  —  x'"  -{■  max'"-'^  +  m  — ; «2^m-2  _^^   ^^^ 

we  suppose    x  =  I,    a  =  1,    we  have, 

m  —  1  m  —  I    m  —  2 

(1  +  !)"•    or   2"^=!  +  m  +  m  — —  +  m  — —  .  — —  +,  &c. 

That  is,  the    sum   of  the   co-ej^cients  of  all   the  terms  of  the  for- 
mula for  the  binomial,  is  equal  to  the  mth  power  of  2. 
Thus,  in  the  particular  case 

(x  +  ay  =  x^  +  5ax^  +  lOa^.r^  +  lOa^x-  -r  5a*x  +  a\ 
the  sum  of  the  co-efficients 

1  +  5+10+10  +  5+1 
is  equal  to  2^  =  32.     In  the  10th  power  developed,  the  sum  of  the 
co-efficients  is  equal  to  2^'^  =  1024. 

Extraction  of  the  Cube  Root  of  Numbers. 

209.  The  cube  or  third  poioer  o^  a  number,  is  the  product  which 
arises  from  multiplying  the  number  twice  by  itself.  The  cube  root, 
or  third  root  of  a  number  is  either  of  three  equal  factors  into 
which  it  may  be  resolved  ;  and  hence,  to  extract  the  cube  root, 
is  to  seek  one  of  these  factors. 

Every  number  which  can  be    resolved    into  three    equal  factors 

that  are   commensurable  with  unity,   is  called  a  perfect  cube;  and 

any  number  which  cannot  be  so  resolved,  is  called  an  imperfect 
cube. 

The  first  ten  numbers  are 

roots,          1,   2,     3,      4,       5,        6,        7,        8,  9,        10; 
cubes,        1,    8,    27,    64,    125,    216,    343,    512,    729,    1000. 

Eeciprocally,  the  numbers  of  the  first  line  are  the  cube  ruois 
of  the  numbers  of  the  second 


208  ELEMENTS    OF    ALGEBRA.  [CHAP.   Vllt. 

We  perceive,  by  inspection,  that  there  are  but  nine  perfect  cubes 
among  all  the  numbers  expressed  by  one,  two,  and  three  figures. 
Every  other  number,  except  the  nine  written  above,  which  can  be 
expressed  by  one,  two,  or  three  figures,  will  be  an  imperfect  cube  ; 
and  hence,  its  cube  root  will  be  expressed  by  a  whole  number, 
plus  an  irrational  number,  as  may  be  shown  by  a  course  of  rea-' 
soning  entirely  similar  to  that  pvu-sued  in  the  latter  part  of  Art.  118. 

210.  Let  us  find  the  difference  between  the  cubes  of  two  con- 
secutive numbers. 

Let  a  and  a  +  1,  be  two  consecutive  whole  numbers;  we  have 

{a+  ly  =  a^  4-  3a2+  3a  +  1  ; 
whence,  (a  +  1)^  —  a^  =  Sa^  -|-  3a  -j-  i. 

That  is,  the  difference  between  the  cubes  of  two  consecutive  whole 
numbers,  is  equal  to  three  times  the  square  of  the  least  number,  plus 
three  times  the  number,  plus  1. 

Thus,  the  difference  between  the  cube  of  90  and  the  cube  of 
89,  is  equal  to 

3  (89)2  +  3  X  89  +  1  =  24031. 

211.  In  order  to  extract  the  cube  root  of  an  entire  number,  we 
will  observe,  that  when  the  figures  expressing  the  number  do  not 
exceed  three,  the  entire  part  of  the  root  is  found  by  comparing 
the  number  with  the  first  nine  perfect  cubes.  For  example,  the 
cube  root  of  27  is  3.  The  cube  root  of  30  is  3,  plus  an  irrational 
number,  less  than  unity.  The  cube  root  of  72  is  4,  plus  an  ir- 
rational number  less  than  unity,  since  72  lies  between  the  perfect 
cubes  64  and  125. 

When  the  number  is  expressed  by  more  than  three  figures,  the 
process  will  be  as  follows.     Let  the  proposed  number  be  103823. 


103  823 

47 

48 

47 

64 

~8 

48 
384 

47 

!  398.23 

329 

192 
2304 

188 

2209 

48 

47 

18432 

15463 

9216 

8836 

110592  103823 

This  number  being  comprised  between  1,000,  which  is  the  cube 


CHAP.  Vin.]  CUBE    ROOT    OF    NUMBERS.  201? 

of  10,  and  1,000,000,  which  is  the  cube  of  100,  its  root  will  be 
expressed  by  two  figures,  or  by  tens  and  units.  Denoting  the  tens 
by  a,  and  the  units  by  h,  we  have  (Art.   198), 

(a  +  hf  =  a?-\-  Zd^b  +  2ab'-  +  l"^. 

Whence  it  follows,  that  the  ube  of  a  number  composed  of  tens 
and  units,  is  made  up  of  four  Jistinct  parts  :  viz.,  the  cube  of  the 
tens,  three  times  the  product  of  the  square  of  the  tens  by  the  units, 
three  times  the  product  of  the  tens  by  the  square  of  the  units,  and 
the  cube  of  the  units. 

Now,  the  cube  of  the  tens,  giving  at  least,  thousands,  the  last 
three  figures  to  the  right  cannot  form  a  part  of  it :  the  cube  of 
tens  must  therefore  be  found  in  the  part  103  which  is  separated 
from  the  last  three  figures.  The  root  of  the  greatest  cube  con- 
tained in  103  being  4,  this  is  the  number  of  tens  in  the  required 
root.  Indeed,  103823  is  evidently  comprised  between  (40)''  or 
64,000,  and  (50)^  or  125,000  ;  hence,  the  required  root  is  com- 
posed of  4  tens,  phis  a  certain  number  of  units  less  than  ten. 

Having  found  the  number  of  tens,  subtract  its  cube,  64,  from 
103,  and  there  remains  39,  to  which  bring  down  the  part  823, 
and  we  have  39823,  which  contains  three  times  the  square  of  the 
tens  by  the  units,  plus  the  two  parts  named  above.  Now,  as  the 
square  of  tens  gives  at  least  hundreds,  it  follows  that  three  times 
the  square  of  the  tens  by  the  units,  must  be  found  in  the  part 
398,  to  the  left  of  23,  which  is  separated  from  it  by  a  point. 
Therefore,  dividing  398  by  48,  which  is  three  times  the  square 
of  the  tens,  the  quotient  8  w^ill  be  the  units  of  the  root,  or  some- 
thing greater,  since  398  hundreds  is  composed  of  three  times  the 
square  of  the  tens  by  the  units,  together  with  the  two  other  parts. 

We  may  ascertain  whether  the  figure  8  is  too  great,  by  forming 
from  the  4  tens  and  8  units  the  three  parts  which  enter  into  39823; 
but  it  is  much  easier  to  cube  48,  as  has  been  done  in  the  above  ta- 
ble. Now,  the  cube  of  48  is  110592,  which  is  greater  than  103823; 
therefore,  8  is  too  great.  By  cubing  47  we  obtain  103823;  hence 
the  proposed  number  is  a  perfect  cube,  and  47  is  its  cube  root. 

Remark  I. — The  units  figure  could  not  be  first  obtained,  because 
the  cube  of  the  units  might  give  tens,  and  even  hundreds,  and  the 
tens  and  hundreds  would  be  confounded  with  those  which  ari^e 
from  other  parts  of  the  cube. 

14 


10  ELEMENTS    OF    ALGEBRA.  [CriAP.  VIII. 

Remark  II. — The  operations  in  the  last  example  have  been  per- 
formed on  but  two  periods.  It  is  plain,  however,  that  the  same 
reasoning  is  equally  applicable  to  larger  numbers  ;  for,  by  chan- 
fj;ing  the  order  of  the  units,  we  do  not  change  the  relation  in 
which  they  stand  to  each  other. 

Thus,  in  the  number  43  725  658,  the  two  periods  43  725,  have 
the  same   relation   to    each  other,  as   in   the    number  43725  ;    and 
hence,  the  methods  pursued  in   the   last    example   are  equally  ap 
plicable  to  larger  numbers. 

212.  Hence,  for  the  extraction  of  the  cube  root  of  numbers,  we 
have  the  following 

RULE. 

I.  Separate  the  given  number  into  periods  of  three  figures  each, 
beginning  at  the  right  hand:  the  left-hand  period  will  often  contain 
less  than  three  places  of  figures. 

II.  Seek  the  greatest  cube  in  the  first  period,  at  the  left,  and  set 
its  root  on  the  right,  after  the  manner  of  a  quotient  in  division.  Sub- 
tract the  cube  of  this  figure  of  the  root  from  the  first  period,  arid  to 
the  remainder  bring  down  the  first  figure  of  the  next  period,  and  call 
ihis  number  the  dividend. 

III.  Take  three  times  the  square  of  the  root  just  found  for  a  di' 
visor,  and  see  how  often  it  is  contained  in  the  dividend,  and  place 
the  quotient  for  a  second  figure  of  the  root.  Then  cube  the  figures 
of  the  root  thus  found,  and  if  their  cube  be  greater  than  the  first 
two  periods  of  the  given  number,  diminish  the  last  figure ;  but  if  it 
be  less,  subtract  it  from  the  first  two  periods,  and  to  the  remaindei 
bring  down  the  first  figure  of  the  next  period,  for  a  new  dividend. 

IV.  Take  three  times  the  square  of  the  while  root  for  a  new  divi- 
sor, and  seek  how  often  it  is  contained  in  the  new  dividend ;  the 
quotient  will  be  the  third  figure  of  the  root.  Cube  the  whole  root., 
and  subtract  the  result  from  the  first  three  periods  of  the  given  num- 
ber, and  proceed  in  a  similar  way  for  all  the  periods. 

Remark. — If   any  of  the   remainders    are    equal    to,  or  exceed. 
three  times  the  square  of  the  root  obtained  plus  three  times  this  root, 
vlus  one,  the  last  figure  of  the  root  is  too  small  and   must  be  aug 
Tiented  by  at  least  unity  (Art.  210). 


CHAP.  VIII.]  EXTRACTION    OF    ROOTS.  311 

EXAMPLES. 


1.  V48228544  =364. 

2.  V27054036008         =  3002. 

3.  ^483249  —  78,   with  a  remainder  8697. 

4.  ^91632508641         =  4508,    with  a  remainder  20644129. 

5.  ^32977340218432  =  32068. 

To  extract  the  v}^  Root  of  a  ivhole  Number. 

213.  The  n'''  root  of  a  number,  is  one  of  the  n  equal  factors 
into  wliich  the  number  may  be  resolved.  If  the  factors  are  com 
mensurable  with  unity,  the  number  is  said  to  be  a  perfect  power, 
if  they  are  not  commensurable  with  unity,  the  number  is  said  to 
be  an  imperfect  power. 

In  order  to  generalize  the  process  for  the  extraction  of  roots, 
we  will  denote  the  proposed  number  by  N,  and  the  degree  of  the 
root  to  be  extracted  by  n.  If  the  number  of  figures  in  N,  does 
not  exceed  n,  the  rational  part  of  the  root  will  be  expressed  by 
a  single  figure :  for  the  n"'  power  of  9  is  the  highest  power  which 
can  be  expressed  by  n  figures. 

Form  the  ?^'*  powers  of  all  the  numbers  from  1  to  9  inclusive. 
Compare  the  given  number  with  these  powers,  and  if  either  of 
them  is  equal  to  the  given  number,  it  will  be  a  perfect  power ;  if 
not,  the  root  of  the  one  next  less  will  be  that  part  of  the  required 
root  wliich  can  be  expressed  by  a  whole  number. 

When  N  contains  more  than  n  figures,  there  will  be  more  than 
one  figure  in  the  root,  which  may  then  be  considered  as  composed 
of  tens  and  units.  Designating  the  tens  by  a,  and  the  units  by  b, 
we  have  (Art.  203), 

n  —  1 

N  =  {a  -\-  b)"  =  a"  +  na-'-^i  +  n  — ; a^-^^  +,   &c. ; 

that  is,  the  proposed  number  contains  the  n^'^  power  of  the  tens, 
plus  n  times  the  product  of  the  n  — l'**  power  of  the  tens  by  the 
units,  plus  a  series  of  other  parts  which  it  is  not  necessary  to 
consider. 

Now,  as  the  n'^  power  of  the  tens,  cannot  give  units  of  an  or- 
der inferior  to  1  followed  by  n  ciphers,  the  last  n  figures  on  the 
right,  cannot  make  a  part  of  it.     They  must  then  be  pointed  ofT 


212  ELEMENTS    OF    ALGEBRA.  [CHAP.   VIII. 

and  the  root  of  the  greatest  n'^  power  contained  in  the  figures  on 
the  left  should  be  extracted  :  this  root  Avill  be  the  tens  of  the  re- 
quired root. 

If  this  part  on  the  left  should  contain  more  than  n  figures,  the 
n  figures  on  the  right  of  it,  must  be  separated  from  the  rest,  and 
the  root  of  the  greatest  n'^  power  contained  in  the  part  on  the  left 
extracted,  and  so   on.     Hence  the  following 

RULE. 

I.  Divide  the  nmnber  N  into  periods  of  n  figures  each,  beginning 
at  the  right  hand ;  extract  the  root  of  the  greatest  n^^  power  con- 
tained in  the  left-hand  period,  and  subtract  the  n'*"  poivcr  of  this 
figure  from  the  left-hand  period. 

II.  Bring  down  to  the  right  of  the  remainder  derived  from  the 
left-hand  period,  the  first  figure  of  the  next  period,  and  call  this 
number  the  dividend. 

III.  Form  the  n  —  1  power  of  the  first  figure  of  the  root,  mul- 
tiply it  by  n,  and  see  how  often  the  product  is  contained  in  the  divi- 
dend :  the  quotient  will  be  the  second  figure  of  the  root,  or  something 
greater. 

IV.  Raise  the  number  thus  formed  to  the  xC-^  power,  then  subtract 
this  result  from  the  two  left-hand  periods,  and  to  the  new  remainder 
bring  down  the  first  figure  of  the  next  period :  then  divide  the  niun- 
ber  thus  formed  by  n  times  the  n  —  1  power  of  the  two  figures  of 
the  root  already  found,  and  continue  this  operation  until  all  the  pe- 
riods are  brought  down. 

EXAMPLES. 

1.  What  is  the  fourth  root  of  531441? 

53  144i    I  27 
2*  =  16 

4  X  23  =  32  I  371 
(27)*  =  531441. 

We  first  divide  off,  from  the  right  hand,  the  period  of  four  figures, 
and  then  find  the  greatest  fourth  root  contained  in  53,  the  first 
period  to  the  left,  which  is  2.  We  next  subtract  the  4th  power 
of  2,  which  is  16,  from  53,  and  to  the  remainder  37  we  bring 
down  the  first  figure  of  the  next  period.'    AVe  then  divide  371  by 


CHAP.   Vllt.]  EXTRACTION    OF    ROOTS.  213 

4  times  the  cube  of  2,  which  gives  11  for  a  quotient :  but  this 
we  know  is  too  large.  By  trying  the  numbers  9  and  8,  we  find 
them  also  too  large:  then  trying  7,  we  find  the  exact  root  to  be  27. 

214.  Remark. — When  the  degree  of  the  root  to  be  extracted  is 
a  multiple  of  two  or  more  numbers,  as  4,  6,  .  .  .  .,  the  root  can  be 
obtained  by  extracting  the  roots  of  more  simple  degrees,  successively. 
To  explain  this,   we   will   remark  that, 

{a^Y  =  a^  X  a^  X  a^  X  a^  =  a3+3+:^+3  ^  a^xi  ^  ai2. 

and  that  in  general  (Art.   13), 

(a"')"  =  a"'  X  a'"  X  a"*  X  a"*  .   .   .    =  a"*^"  : 

hence,  the  n***  power  of  the  m"*  power  of  a  number,  is  equal  to  ike 
mn'*"  power  of  this  number. 

Let  us  see  if  the  reciprocal  of  this   is   also  true. 


Let  \J  -yf  a  :=  a'' ; 

then  raising  both  members  to  the  n"^  power,  we  have,  from  the 
definition  of  the  n^^  root, 

V  a  =  «'«  ; 

and  by  raising  both  members  of  the  last  equation  to  the  m'*  power 

a  =  (a'")'"  =  o/'"". 

Extracting  the   mn"^  root  of  the  last  equation,  we  have 

mil  I  , 

yj  a  ^=  a  \ 

"    A 
and  hence, 

since  each  is  equal  to  a' .  Therefore,  the  n'**  root  of  the  m""  root 
of  any  number,  is  equal  to  the  mn''>  root  of  that  number.  And  in 
a  similar  manner,  it  might  be  proved  that 


n  /  mn  / 

y  a  =     Y  a. 
By  this  method  we  find  that 


1.  V  256  =  y -/  256  =  ^/l6  =  4. 

2      V  2985984  =  y  V  2985984  =  \/l728  =  12. 


214  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIII 

3  V  1771561   =  \J  ^fvn\bE\  =  1 1 . 

4.  V  1679616   =  Vl296  =  -^^1296  =  6. 

Remark. — Although  the  successive  roots  may  be  extracted  in 
any  order  whatever,  it  is  better  to  extract  the  roots  of  the  lowest 
degree  first,  for  then  the  extraction  of  the  roots  of  the  higher 
degrees,  which  is  a  more  complicated  operation,  is  effected  upon 
numbers   containing  fewer  figures   than  the   proposed  number 

Extraction  of  Roots  by  Ajjproximation. 

215.  When  it  is  required  to  extract  the  n"^  root  of  a  number 
which  is  not  a  perfect  power,  the  method  already  explained,  will 
give  only  the  entire  part  of  the  root,  or  the  root  to  within  unity. 
As  to  the  number  which  is  to  be  added,  in  order  to  complete  the 
root,  it  cannot  be  obtained  exactly,  but  we  can  approximate  to  it 
as   near  as   we   please. 

Let  it  be  required  to  extract  the  n'^  root  of  the  whole  number 

a  to  within  a  fraction   — ;    that  is,   so  near,   that    the   error   shall 
P 

be  less  than    — . 
P 
We  will  observe  that  we   can  write 

«»" 

If  we  denote  by  r,  the  root  of  op"  to  within  unity,  the  number 

a  X  p"                                                                           ^"              (^  "t"  ^Y 
i—  =  a,     will  be   comprehended   between    —    and ; 

pn  '  ^  pn  pn 

therefore  the   "■</~a    will  be  comprised  between  the  two  numbers 

r  r  -1-  1  1 

—  and ;    and  consequently,  their  difference    —    will   be 

p  P  ■  P 

r 

weater  than  the  difference  between  —  and  the  true  root.     Hence, 
b  p 

r  .  .  1 

—  is  the   required  root  to  within   the   fraction   — . 

P  P 

Hence,  to  extract  the  n^^  root  of  a  whole  number  to  within  a  frac- 

lion    — ,    multiply  the    number   by  p" ;    extract   the    n*'^    root    of  the 

P 
oroduct  to  within  unity,  and  divide  the  result  by  p. 


CHAP  VIII.]  EXTRACTION    OF    ROOTS.  215 

216.  Again,  suppose  it  is  required  to  extract  the  n'^  root  of  the 
fraction   -—. 

0 

Multiply  each  term  of  the  fraction  by- 
fir         ai"~^ 

b"~'^,    and  it  becomes    -—  =  — , . 

b  b" 

Let  r  denote   the  n'^  root  of  ab"~^,  to   within   unity ; 

— ; =  -— ,    will  be   comprised  between    —-    and    ^^ — - — —  ; 

b""  b  ^  b"  b" 

T  CI 

and  consequently,    -—    will  be  the   n'^  root  of  — -,    to   within   the 

0  0 

fraction    -— . 

0 

Therefore,  after  having  made  the  denominator  of  the  fraction  a 
■perfect  power  of  the  n''^  degree,  extract  the  r\}^  root  of  the  numerator, 
to  within  unity,  and  divide  the  result  by  the  root  of  the  new  de- 
nominator. 

When  a  greater  degree  of  exactness  is  required  than  that  in- 
dicated by   -— ,    extract  the    root  of  ai"~^  to  within  any  fraction 

0 

1  .  r'  .  r^     . 

— ;    and  designate  this  root  by   — .     Now,  since    —    is  the  root 

P  P  P 

1       .  r'     . 

of  the   numerator    to  within    — ,   it   follows,  that   -r—   is    the    true 

P  bp 

root  of  the   fraction  to   within    -7-. 

bp 

EXAMPLES. 

1.  Suppose  it  were  required  to  extract  the  cube  root  of  15,  to 
within    -;-.     We   have 

1  ^ 

15  X  123  —  15  X  1728  =  25920. 
Now  the   cube  root  of  25920,  to  within   unity,  is   29  ;   hence 
the   required  root  is, 

29  5 

—  =z2— . 
12  12 

2.  Extract  the  cube  root  of  47,   to   within    — . 

20 

We  have 

47  X  203  =  47  X  8000  =  376000. 


216  ELEMExVTS    OF    ALGEBRA.  [CHAP.  VIII. 

Now  the  cube  root  of  376000,  to   within  unity,   is   72 ;    hence 

V  47  =  —  =  3 — ,    to  within    — . 
^  20  20  20 

3.  Find  the  vahie  of  ^25   to  within  0.001. 

To  do  this,  multiply  25  by  the  cube  of  1000,  or  1000000000, 
which  gives  25000000000.  Now,  the  cube  root  of  this  number, 
is   2920 ;   hence 

3 


.  25  =  2.920    to  within  0.001. 

217.  Remark. — In  general,  in  order  to  extract  the  cube  root  of  a 
whole  number  to  within  a  given  decimal  fraction,  annex  three  times  as 
many  ciphers  to  the  number,  as  there  are  decimal  places  in  the  required 
root ;  extract  the  cube  root  of  the  number  thus  formed  to  withiri 
unity,  and  point  off  from  the  right  of  this  root  the  required  numhei 
of  decimals. 

218.  We  will  now  explain  the  method  of  extracting  the  cube 
root  of  a  decimal  fraction.  Suppose  it  is  required  to  extract  the 
cube  root  of  3.1415. 

Since  the  denominator,  10000,  of  this  fraction,  is  not  a  per- 
fect cube,  make  it  one,  by  multiplying  it  by  IQO  ;  this  is  equivalent 
to  annexing  two  ciphers  to  the  proposed  decimal,  which  then  be- 
comes, 3.141500.  Extract  the  cube  root  of  3141500,  that  is,  of 
the  number  considered  independent  of  the  comma,  to  within  unity ; 
this  gives  146.  Then  divide  by  100,  or  \AoOOOOO,  and  we  find 
V  3.1415  =  1.46  to  within  0.01. 

Hence,  to  extract  the  cube  root  of  a  decimal  number,  we  have 
the  following 

RULE. 

Annex  ciphers  to  the  decimal  part,  if  necessary,  until  it  can  be 
divided  into  exact  periods  of  three  figures  each,  observing  that  the 
number  of  periods  must  be  made  equal  to  the  number  of  decimal 
places  required  in  the  root.  Then,  extract  the  root  as  in  entire  num- 
bers, and  point  off  as  many  places  for  decimals  as  there  are  periods 
11  the  decimal  part  of  the  number. 

To  extract  the  cube  root  of  a  \'ulgar  fraction  to  within  a  given 
decimal  fraction,  the  most  simple  method  is  to  reduce  the  pro- 
posed fraction   to   a  decimal  fraction,   continuing  the   operation  until 


CHAP.   VIII.]  EXTRACTION    OF    ROOTS.  217 

the  nninher  of  decimal  places  is  equal  to  three  times  the  numher  re- 
quired in  the  root.  The  question  is  then  reduced  to  extracting  the 
cube  root  of  a  decimal  fraction. 

219.  Suppose  it  is  required  to  find  the  sixth  root  of  23,  to 
within  0.01. 

Applying  the  ride  of  Art.  215  to  this  example,  we  multiply  23 
by  lOO*",  or  annex  twelve  ciphers  to  23  ;  then  extract  the  sixth  root 
of  the  number  thus  formed  to  within  unity,  and  divide  thi.s  root 
by  100,  or  point  off  two  decimals  on  the  right. 

We  thus  find  that,    ^23  =  1.68,  to  within  0.01. 

EXAMPLES. 

1.  Find  the  \^ AlZ    to  within  jV*  -^'"'-  ^4- 

2.  Find  the  ^Tg"   to  within  .0001.  Ans.  4.2908. 

3.  Find  the  \f~Vi     to  within  .01.  Ans.   1.53. 

4.  Find  the  ^3-004 15    to  within  .0001.  Ans.  1.4429. 

5.  Find  the  Vo.OOlOl    to  within  .01.  Ans.  0.10. 

6.  Find  the  %/  ^^    to  within  .001.  Ans.  0.824. 

Extraction  of  Roots  of  Algebraic   Quantities. 

220.  Before  extracting  the  root  of  an  algebraic  quantity,  let  us 
see  in   what  manner   any  power  of  it   may  be   formed. 

Let  it  be  required  to  form  the  fifth  power  of  2a%'^.  We  have 
{2a'^h'^y  =  2a%'^  X  2aW  X  2aW  X  2aW  X  2aW, 
from  which  it  follows,  1st.  That  the  co-efficient  2  must  be  mul- 
tiplied by  itself  four  times,  or  raised  to  the  5th  power.  2d.  That 
each  of  the  exponents  of  the  letters  must  be  added  to  itself  four 
times,  or  multiplied  by  5. 

Hence,  {2a%'^f  =  2^  .  a3x5^,2X5  _  32ai5jio. 

In  like  manner,      {QaWcy  =  8^  .  a'^^^¥>^^c'^  =  512a^^c^. 

Therefore,  in  order  to  raise  a  monomial  to  any  power,  raise 
the  co-efficient  to  that  power,  and  multiply  the  exponent  of  each  of 
(he  letters  hy  the  exponent  of  the  power. 

Hence,  to   extract  any  root   of  a  monomial, 

1st.  Extract  the  root  of  the  co-efficient  and  divide  the  exponent 
of  each  letter  hi/  the  index  of  the  root.     2d.    To  the   root   of  the  co 


218  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIII. 

efficient  annex   each  letter  with  its  new  exponent,  and  the  result  will 
be  the  required  root.     Thus, 


y^Aa^^c^  =  Aa^bc"^  ;       \/\Qa%^\^  =  laWc. 

From  this  rule  we  perceive,  that  in  order  that  a  monomial  may 
De  a  perfect  power,  1st,  its  co-efficient  must  be  a  perfect  power; 
and  2d,  the  exponent  of  each  letter  must  be  divisible  by  the  in- 
dex of  the  root  to  be  extracted.  It  will  be  shown  hereafter,  how 
the  expression  for  the  root  of  a  quantity,  which  is  not  a  perfect 
power,  is  reduced  to  its  simplest  terms. 

221.  Hitherto,  in  finding  the  power  of  a  monomial,  we  have 
paid  no  attention  to  the  sign  with  which  the  monomial  may  be 
affected.  It  has  already  been  shown,  that  whatever  be  the  sign 
of  a  monomial,  its  square  is  always  positive. 

Let  n  be  any  whole  number ;  then,  every  power  of  an  even 
degree,  as  2n,  can  be  considered  as  the  n'^  power  of  the  square ; 
that  is,   {a^Y  =  a^n. 

Hence,  it  follows,  that  every  power  of  an  even  degree,  will  be  es- 
sentially positive,  whether  the  quantity  itself  be  positive  or  negative. 

Thus,  (d=  2a^Pcy  =  +  16a^^^c\ 

Again,  as  every  power  of  an  uneven  degree,  2n  -\-  1,  is  but  the 
product  of  the  power  of  an  even  degree,  2«,  by  the  first  power ; 
it  follows  that,  every  power  of  an  uneven  degree,  of  a  monomial,  is 
affected  with  the  same  sign  as  the  monomial  itself. 

Hence,     {-{- Aa^y  ^  +  64a%^  ;    and    {— 4a^by  =  —  64a^^ 
From  the  preceding  reasonings,  we  conclude, 
1st.    That  when  the   degree  of  the  root  of  a  monomial   is    uneven, 
the  root  will  be  affected  with  the  same  sign  as  the  monomial. 
Hence, 

V+  8a3  =  +  2a  ;     \/ -  8a^  =  -2a;     %/ -  32a^%^  =  -  20^. 

2d.  When  the  degree  of  the  root  is  even,  and  the  monomial  a  posi- 
tive quantity,  the   root  is  affected  either  with  the  sign   -\-   or  — . 

Thus,  V81a*5i2=  ±3aJ3;       \/&l'a^  =  ±  2a^ . 

c5d.  When  the  degree  of  the  root  is  even,  and  the  monomial  nega- 
tive, the  root  is  impossible ;  for,  there  is  no  quantity  which,    beint 


CHAP.   VIII.]  EXTRACTION    OF    ROOTS.  219 

raised  to  a  power  of  an  even  degree,  will   give    a  negative    result. 
Therefore, 

4/  6/       r        8/ 

V  —  «.       V  —  0,       W  —  c, 

are  symbols  of  operation  which  it  is  impossible  to  execute.     They 
are  imaginary  expressions  (Art.   126),  like 

V— a,       y/—b, 

EXAMPLES. 

1.  \Vhat  is  the  cube  root  of  Sa%^c^'^'f.  Ans.  2a'^bc* 

2.  What  is  the  4th  root  of   81  a^Pc^^  1  Ans.  3a/;V. 

3.  What  is  the  5th  root  of    —  32a^c^'^d^^  ?  Ans.   —  2ac^d^. 

4.  What  is  the  cube  root  of    —  125a^^c^  ?  Ans.   —  baWc. 

Extraction  of  Roots  of  Polijnomiah. 

222.  Let  us   first   examine   the   law   of  formation  of  any  power 
of  a  polynomial.     To  begin  with  a  simple  example,  let  us  develop 
(a  +  y  +  zy. 
If  we  place   y  -\-  z  =z  u,   we  shall  have, 

(a  +  w)3  =  a3  +  3a^-u  +  3au^  +  u^  ; 
or  by  replacing  u  by  its  value,  y  -^  z, 

{a  +  y  +  zY  =  a3  +  Sa^  {y  +  z)  +  3a  [y  +  zf  +  (y  +  zf ; 
or  performing  the  operations  indicated, 

(a  +  y  +  2)3  =  a3  +  3a2y  +  3a^z  +  3«y2  +  Qayz  +  3az^  +  y3+  3y'^z 

+  3yz^  +  ^^• 
When  the  polynomial  is  composed  of  more  than  three  terms,  as 
a  -}-  y  -\-  z  -{-  X  .  .  .  .  p,   let,  as   before,  u  =   the   sum  of  all  the 
terms  after  the    first.     Then,    a  -\-  u   will   be    equal   to   the    given 
polynomial,  and 

(a  +  w)3  =  a3  -f  3dhi  -|-  3mfi  +  rfi  ; 
from  which  we  see,  that  the  cube  of  any  polynornial  is  equal  to  the 
cube  of  the  first  term,  plus  three  times   the  square   of  the  first  term 
multiplied  by  each  of  the  remaining  terms,  plus  other  terms. 

If  u  does  not  contain  a,  it  is  plain  that  the  exponent  of  a  in 
each  term,  as  a^,  3a'^u,  &c.,  will  be  greater  than  in  any  of  the 
following  terms  ;  and  hence,  every  term  will  be  irreducible  with  the 
terms  which  precede  or  follovj  it. 


220  ELEMKNTS    OF    ALGEBRA.  [CHAP.  VIII. 

If  u  contains  a,  as  in  the  polynomial 

a^  -{-  ax  -\-  b,    where       u  =  ax  -{-  b, 

the  terms  will  still  be  irreducible  with  each  other,  provided  we 
arrange  the  polynomial  with  reference  to  the  letter  a.  For,  if  the 
given  polynomial  be  arranged  with  reference  to  a,  the  exponent 
of  a  in  the  first  term  will  be  greater  than  the  exponent  of  a  in 
XI :  hence,  its  cube  will  contain  a  with  a  greater  exponent  than 
will  result  from  multiplying  its  square  by  u.  Also,  the  co-efficient 
of  u  multiplied  by  the  first  term  of  w,  will  contain  a  to  a  higher 
power  than  any  of  the  following  terms  of  the  development,  and 
hence,  will  be  irreducible  with  them  ;  and  the  same  may  be  shown 
for  the  subsequent  terms. 

In  order  to  extract  any  root  of  a  polynomial,  we  will  first  ex- 
plain the  method  of  extracting  the  cube  root.  It  will  then  be 
easy  to  generalize  this  method,  and  apply  it  to  the  case  of  any 
root  whatever. 

Let  N  be  any  polynomial,  and  R  its  cube  root.  Suppose  the 
two  polynomials  to  be  arranged  with  reference  to  some  letter,  as 
a,  for  example.  It  results  from  the  law  of  formation  of  the  cube 
of  a  polynomial  (Art.  222),  that  in  the  cube  of  R,  the  cube  of 
the  first  term,  and  three  times  the  square  of  the  first  term  by  the 
second,  cannot  be  reduced  with  each  other,  nor  with  any  of  the 
following  terms. 

Hence,  the  cube  root  of  that  term  of  N  which  contains  a,  af- 
fected with  the  highest  exponent,  will  be  the  first  term  of  R ;  and 
the  second  term  of  R  will  be  found  by  dividing  the  second  term 
of  N  by  three  times  the  square  of  the  first  term  of  R. 

By  examining  the  development  of  the  trinomial  a  -\-  y  -\-  z,  we 
see,  that  if  we  form  the  cube  of  the  two  terms  of  the  root  found 
as  above,  and  subtract  it  from  N,  and  then  divide  the  first  term 
of  the  remainder  by  3  times  the  square  of  the  first  term  of  i2, 
the  quotient  will  be  the  third  term  of  the  root.  Therefore,  having 
arranged  the  terms  of  N,  with  reference  to  any  letter,  we  have, 
for  the   extraction  of  the   cube  root,  the   following 

RULE. 

I.  Extract  the  cube  rout  of  the  first  term. 

II.  Divide  the  second  term  of  1^  by  three  times  the  square  of  the 
first  term  o/"  R  ;  the  quotient  will  be  the  second  term  of  R. 


CHAP.   VIII.l  EXTRACTION    OF    ROOTS.  221 

[II.  Having  found  the  first  two  terms  of  R,  form  the  cube  of  this 
binomial  and  subtract  it  from  N  ;  after  ivhicJi,  divide  the  first  term 
of  the  remainder  by  three  times  the  square  of  the  first  term  of  R  : 
the  quotient  will  be  the  third  term  of  R. 

IV.  Cube  the  three  terms  of  the  root  found,  and  subtract  the  cube 
from  N  :  then  divide  the  first  term  of  the  remainder  by  the  divisor 
ilrcadij  used,  and  the  quotient  will  be  the  fourth  term  of  the  root: 
the  remaining  terms,  if  there  are  any,  may  be  found  in  a  similar 
manner. 

EXAMPLES. 

1.  Extract  the  cube  root  of  x^—6x^-\-\5x'^—20x'^-\-  Ibx^—Sx+l. 

a;"— 6j;H  15a;*  -  20x3  +  15^2  —  6j;  + 1  \yZ_2x-{-\ 
(a;2  —  2a7)3  =  ar6—6a;5+ 12^-4—    Sx"^  3a;* 

1st  rem.  Sx*  —  12a;3 +,  &c. 

(a:2— 2a;+l)3  =  a;<5— 6a;5  +  15x*  — 20a;3  +  15x2— 6x  +  1. 

In  this  example,  we  first  extract  the  cube  root  of  x^,  which 
gives  a;2,  for  the  first  term  of  the  root.  Squaring  x"^,  and  multi- 
plying by  3,  we  obtain  the  divisor  3a;* :  this  is  contained  in  the 
second  term  —  &x^,  —  2a;  times.  Then  cubing  the  root,  and  sub- 
tracting, we  find  that  the  first  term  of  the  remainder  3a;*,  contains 
the  divisor  once.  Cubing  the  whole  root,  we  find  the  cube  equal 
to  the  given  polynomial.  Hence,  a?2  — 2a;  +  1,  is  the  exact  cube 
root. 

2.  Find  the  cube  root  of 

a^6  +  6a;5  —  40x3  _|_  qq^  _  54 

3.  Find  the*  cube  root  of 

8x6  _  12x5  +  30x*  —  25a;3  +  30x2  _  12a?  +  8 

223.  The  rule  for  the  extraction  of  the  cube  root  is  easily  ex- 
tended to  a  root  with  a  higher  index.     For, 

Let  a  ■\-  b  -\-  c  -\-   .  .  .  f    be  any  poljmomial. 

Let    s  =  the   sum  of  all  the   terms   after  the   first. 
Then   a  -{-  s  ■=    the  given  polynomial  ;    and 

(a  +  sy  =  a"  -f  na"~'^s  +     other  terms. 
That  is,  the  n*''  power  of  a  polynomial,  is  equal  to  the  n^^  power 
of  the  first   term,  plus   n    times   the  first  term   raised  to   the  power 


222  ELEMENTS    OF    ALGEBRA.  ICHAP.   VIII. 

n  —  1,  multiplied  by  each    of  the   remaining   terms,    -\-    other    terms 
of  the  development. 

Hence,  we  see,  that  the  rule  for  the  cube  root  will  become  the 
rule   for  the  n"^  root,  by  first    extracting   the  n"^  root  of  the   first 
term,  taking  for  a  divisor  n  times  this  root  raised  to  the    n  —  1 
power,  and  raising  the   partial  roots  to  the  n^'^  power,  instead  of' 
to  the  cube. 

EXAMPLES. 

1.  Extract  the  4th  root  of 

16a*  —  QQa^x  +  2\Qa^x'^  —  216«x3  +  81x*. 

16a*—  96a3a;+216a2a:2— 216ax3+81a;*      2a  — 3a; 
(2a  —  3.r)*  =  16a*-  96a3a;4-216a2a;2— 216ax3  +  81a;*    4x(2a)3  =  32a3 

We  first  extract  the  4th  root  of  16a*,  which  is  2a.  We  then 
raise  2a  to  the  third  power,  and  multiply  by  4,  the  index  of  the 
root ;  this  gives  the  divisor  22a?.  This  divisor  is  contained  in 
the  second  term  —  96a%,  —  Sa?  times,  which  is  the  second  term 
of  the  root.  Raising  the  whole  root  to  the  4th  power,  we  find 
the  power  equal  to  the  given  polynomial. 

2.  What  is  the  4th  root  of  the  polynomial, 

SlaV  +  16^*i*  —  96a2c53d3  _  2\QaH'^bd  +  216a*c2i2(Z2, 

3.  Find  the  5th  root  of 

32x5  _  80a;*  +  80a:3  —  AOx^  +  lOx  —  1. 

Calculus  of  Radicals. 

224.  When  the  monomial  or  polynomial  whose  root  is  to  be  ex- 
tracted, cannot  be  resolved  into  as  many  equal  and  rational  fac- 
tors as  there  are  units  in  the  index  of  the  root,  it  is  said  to  be 
an  imperfect  power.  The  root  is  then  indicated  by  placing  the 
quantity  under  the  radical  sign,  and  writing  over  it  at  the  left 
hand,  the  index  of  the  root.  Thus,  the  fourth  root  of  3ai2  -}-  9ac*, 
is  written 

V3a62  +  9ac5. 

The  index  of  the  root  is  also  called  the  index  of  the  radical. 
It  is  plain  that  a  monomial  will  be  a  perfect  power,  when  the 
numerical  co-efficient  is  a  perfect  poioer,  and  the  exponent  of  each 
Utter  exactly  dvvisihle   hy  the  index  of  the  root. 


CHAP.   VIU.]  CALCULUS    OF    RADICALS.  223 

By  the    definition  of  a  root   (Art.  213),  we  have 
(\/abc  .  .  .  y^  —  abc  .  .  .  ; 
and  by  the   rule   for  the   raising  of  powers, 

and  since  the  n'^  powers  are  equal,  the  quantities  themselves  are 
equal :    hence, 

Y  aic  .  .  .  .   =\/  a    x\/  b    Xyc... 

that  is,  the  n'**  root  of  the  product  of  any  number  of  factors,  is  equal 
to  the  product  of  their  n'**  roots. 

1.  Let   us    apply  the    above   principle    in    reducing   to    its    sim 
plest  form  the  imperfect  power,    y  54tt*6'^c2.     We  have 


ybAa^bH"^  =  \/21aW  X  \/2ac^  =  Sab  y^Zac"^. 

2.  In  like  manner, 

\f^  =  2  y^;     and    y48a^b^c^  —  2ab^c  yi.ac^  ; 

3.  Also, 

Vl92a^ki2=  y64aV2  x  yTd>  =  2ac^  y^ab. 

In  the  expressions,   Sab  -y/2ac2,    2  y  a^,   2ah'^c  ySac^,   each  quan- 
tity placed  before  the  radical,  is  called  a  co-ejjicient  of  the  radical. 

225.  The  rule   of  Art.  214   gives  rise  to   another  kind  of  sim- 
plification. 

.  6  / 

Take,    for    example,  the   radical    expression,    y  Aa^ ;    from    this 
rule,  we  have 


''4a2 


\J y  \a^. 


and  as  the  quantity   affected  with  the  radical  of  the   second   de- 
gree, y ,  is  a  perfect  square,  its  root  can  be  extracted :  hence. 

y~\^^  =  y2a. 

In  like  manner, 

yS^aW  =  \J y  SGaH'^  =  yOab. 
In  general, 

m  /■ 


224  ELEMENTS    OF    ALGEBRA.  [CHAP,   VIU 

that  is,  when  the  index  of  a  radical  is  a  multiple  of  any  number 
w,  and  the  quantity  under  the  radical  sign  is  an  exact  n^''-  power, 
we  can,  without  changing  the  value  of  the  radical,  divide  its  index 
by  n,  and  extract  the  n*^**   root  of  the   quantity  under  the  sign. 

This  proposition  is  the  inverse  of  another,  not  less  important ; 
VIZ.,  the  index  of  a  radical  may  he  multiplied  hy  any  number,  pro- 
vided we  raise  the  quantity  under  the  sign  to  a  power  of  which  this 
'number  is  the  exponent. 

For,  since  a  is  the  same  thing  as    y  a",  we  have, 


a   =  -w  Y  a" 


226.  This  last  principle  serves  to  reduce  two  or  more  radicals 
to  a  common  index. 

For  example,  let  it  be  required  to  reduce  the  two  radicals 

^/2a    and     y  (c  +  b) 
to  the  same  index. 

By  multiplying  the  index  of  the  first  by  4,  the  index  of  the  sec- 
ond, and  raising  the  quantity  2a  to  the  fourth  power  ;  then  multi- 
plying the  index  of  the  second  by  3,  the  index  of  the  first,  and 
cubing  a  -i-  b,  the  value  of  neither  radical  will  be  changed,  and 
the  expressions  will  become 

V^  =  V  2*a*  =  V  16a4  ;     and    V(7+l)  =  V  («  +  ^)^- 

Hence,  to  reduce  radicals  to  a  common  index,  we  have  the  fol- 
lowing 

RULE. 

Midtiply  the  index  of  each  radical  by  the  product  of  the  indices 
of  all  the  other  radicals,  and  raise  the  quantity  under  each  radical 
sign  to  a  power  denoted  by  this  product. 

This  rule,  which  is  analogous  to  that  given  for  the  reduction 
of  fractions  to  a  common  denominator,  is  susceptible  of  similar 
modifications. 

For  example,  reduce  the  radicals 

to  the  same  index. 

Since  24  is  the  least  common  multiple  of  the  indices  4,  6,  and 
8,  it  is  only  necessary  to  m\iUiply  the    first   by  G,  the   second  by 


CHAP.   VIII. J  CALCULUS    OF    RADICALS.  225 

4,  and  the  third  by  3,  and  to  raise  the  quantities  under  each  rad- 
ical sign  to  the  6th,  4tli,  and  3d  powers  respectively,  which 
ffives 


In   applying  the   above   rules   to  numerical   examples,  beginners 
very  often   make  mistakes  similar  to   the   following :  viz.,  in  redii 

cing  the  radicals  y  2  and  y  3  to  a  common  index,  after  having 
multiplied  the  index  of  the  first,  by  tliat  of  the  second,  and  the 
index  of  the  second  by  that  of  the  first,  then,  instead  of  multiply- 
ing the  f.xponent  of  the  quantity  under  the  first  sign  by  2,  and  the 
exponent  of  that  under  the  second  by  3,  they  often  multiply  the 
quantity  under  the  first  sign  by  2,  and  the  quantity  under  the  sec- 
ond by  3.     Thus,  they  would  have 

yr^  y-Z  X  2  =  y^,  and  V^'=  V3~xl  =  Vq" 
Whereas,  they  should  have,  by  the  foregoing  rule, 

y~2  =  V  (2)2  =  \/T,  and  y/T  =  %/~{^  =  \f^ 
Reduce     y  2,     y  4,     y  ?„    to  the  same  index. 

Addition  and  Suhtraction  of  Radicals. 

227.  Two  radicals  are  similar,  when  they  have  the  same  index., 
md  the  same  quantity    under  the  sign.     Thus, 

3  y  ai     and     7  -y/ ab  ;     as  also,     3a-  yi^'     ^^^     ^^^  y^ 
are  similar  radicals. 

In  order  to  add  or  subtract  similar  radicals,  add  or  subtract  their 
co-pfficients,  and  to  the  sum  or  difference  annex  the  common  radical 

Thus, 

3  VT  +  2  \/T=  5  VT";    also,    3  %/T-  2  VT=  V^ 
Again,  Ba^/T  ±  2c  \/T=  (3a  ±  2c)  \fT. 

Dissimilar  radicals   may  sometimes  be  reduced  to   similar  radi- 
cals, by  the  rules  of  Arts.  224  and  225.     For  example, 

i       V"48'aZ^  +  b  y/lba  =  \b  y,f^ia  +  55  yjYa  =  9b  y/da. 
[15  J 


22G  ELEJIENTS    OF    ALGEBRA.  [CHAP.  VIIl 

2.  V8tt3i+  16a*  —  V^*  +  2a/^3  =  2a  V^  +  2a  -  i  V^  +  2a; 

=  (2a  —  b)  yb  +  2a. 

3.  3  V^+2V2^=  3V^  +  2y2^=  5^2^- 

When  the  radicals  are  dissimilar  and  irreducible,  they  can  only 
be  added  or  subtracted,  by  means  of  the  signs   +   or  — . 

Multiplication  mid  Divisio7i. 

228.  We  will  suppose  that  the  radicals  haA'^e  been  reduced  to  a 
common  index. 

Let  it  be   required  to  multiply    y  o    by    y  Z>. 
If  we  denote  the  product  by  P,  we  have 

and  by  raising  both  members  to  the  n"^  power, 

(V^)"  X  ( VT)"  =  abzzz  P"  ; 
and  by  extracting  the  n^^  root, 

V^  X  "y/T=  P  =  \/^; 

that  is,  the  product  of  the  n"'  roots  of  two  quantities,  is  equal  to  the 
n'"  root  of  their  product. 

Let  it  be  required  to  divide    y  a     by    V  ^* 

If  we  designate  the  quotient  by    Q,  we   have 

Q; 

b 

and  by  raising  both  members  to  the   n"*  power, 
"«  / —     —       —  Vt   » 

(V  by     b 

and  by  extracting  the  n"'  root, 

that  is,  the  quotient  of  the  n"^   roots  of  two  quantities,  ts  equal  to 
the  n'**  root  of  their  quotient. 


CHAP.   VIII.]  CALCULUS    OF    RADICALS.  22'/ 

Therefore,  for  the  muhiplication  and  division  of  radicals,  we 
have  the  following 

RULE. 

I.  Reduce  the  radicals  to  a  common  index. 

II.  If  the  radicals  have  co-ejicients,  frst  midtiphj  or  divide  them 
separately. 

III.  Multiply  or  divide  the  quantities  under  the  radical  sign  by 
each  other,  and  prejix  to  the  product  or  quotient,  the  common  radi' 
cal  sign.  , 

EXAMPLES. 

1.  The  product  of 

^        c  *  a  ^  cd 

_       6a2(a2  +  ^>2) 

2  The  product  of 

3a  V^  X  2h  V  W^c  =  Gab  \/  32a*c  =  12a^^/2c. 

3  The  quotient  of 

4.  The  product  of 

3a  VTx  55  Vic  =  15aJ  X  ^\/8b*cK 

5.  Multiply     y/~2x  V~^  by    V  y  X  V  y 

Ans.  ^YT. 

6    Multiply     2  ^15    by    3  ^10. 

Ans.  6  V337500. 
5   r 

3 


7.  Multiply     4\/-|-    by    2\/— . 


10 

.Ans.  8 


V     256 


228  ELEMENTS    OF    ALGEBRA.  [CHAP.  Vlil 

•'       „   ,  2  v^  X  VT         .     , 

8.  Keduce     — ,   . — ^-?=^    to  its  lowest  terms. 

Ans.  4^V^28& 


o    T,    1  .  /V  i  X  2  VY         .     , 

9.  Reduce     V      ..  ^ 7^-=    to  its  lowest  terms. 

4  V  2  X  V  3 

10.  Multiply     ^2,    \fT,    and    i/T'    together. 

yln^.  V648000. 
7    rr     3    A—  

1 1 .  Multiply    V  -TT'      V  "o"'    ^^^    *  V   65    together. 

V  27 

/ —  / —  / —  / — 

12.  Multiply     (4\/y+5Vl)     by    (i/I- +  2  \/i-). 

43       13     ^- 

13.  Divide     yVy    ^'X    (>^+ ^  V  y) 


14.  Divide  1  by   VT+  V^ 


.Ani'. 


a  —  b 
15.  Divide    \f^  +  V^i"   by    y^  -  \/T. 


^ra*. 


Powers  and  Roots  of  Radicals. 
229.  By  raising  %/"«"  to  the  ti"'  power,  Ave  have 

(V7)"  =  V7x  V«  X  V^-  •  •  =  v^. 

by  the  rule  just  given  for  the    uiultiplication  of  radicals.     Hence, 
for  raising  a  radical  to  any  power,  we  have  the  following 


CHAP.  VIII.]  CALCULUS    OF    RADICALS.  229 

•  RULE. 

Raise  the  quantity  under  the  sign  to  the  given  power,  and  affect 
the  result  with  the  radical  sign,  having  the  primitive  index.  If  it 
has  a  co-efficient,  Jirst  raise  it  to  the  given  power. 

EXAMPLES. 

1 .  ( V  4^)^  =  V{4aT  =  Vl6«'  =  2«  \/lfi. 

2.  (3  y2af  =  3-^ .  \f(2d)'  =  243  ^32^  =  4S6a  \/ Aa^. 

When  the  index  of  the  radical  is  a  muhiple  of  the  power  lo 
which  it  is  to  be  raised,  the  result  can  be  simplified. 

For,  Y  2a  =  v/ Y  2a  (Art.  214):  hence,  in  order  to  square 
Y2a,  we  have  only  to  omit  the  first  radical,  which  gives 


Again,  to   square     -y/ 3(5>,  we  have    %/'ob  =  \/%/^:    hence, 

Consequently,  when  the  index  of  the  radical  is  divisible  hij  the 
exponent  of  the  power  to  which  it  is  to  be  raised,  perform  the  di- 
vision, leaving  the  quantity  under  the  radical  sign  unchanged. 

Let  it  be  required  to  extract  the  W^  root  of  the  radical  Y  a. 
We  have  (Art.  214), 

m     I     '  _ 

•    /  "  /  "•«  /~ 

V/  Y  a   =     Y  o. 

Hence,  to  extract  the  root  of  a  radical,  multiply  the  index  of  the 
radical  by  the  index  of  the  root  to  he  extracted,  leaving  the  quan- 
tity under  the  sign  unchanged. 

This  rule  is  nothing  more  than  the  principle  of  Art.  214,  enun- 
ciated in  an  inverse  order. 


1.    yVic^^'V^;  and    y\/Tc  =  %/Tc. 


When  the  quantity  under  the  radical  is  a  perfect  power,  of  the 
degree  of  either  of  the  roots  to  be  extracted,  the  result  can  be 
siinplined. 


230  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIII 

Thus, 


larmer,      y : 


In  like  m 

230.  The  rules  just  demonstrated  for  the  calculus  of  radicals 
depend  upon  the  fact,  that  the  n^'^  root  of  the  product  of  several 
factors,  is  equal  to  the  product  of  the  n'^  roots  of  these  factors 
This,  however,  has  been  proved  on  the  supposition  that,  wlien  the 
powers  of  the  same  degree  of  two  expressions  are  equal,  the  expres- 
sions themselves  are  also  equal.  Now,  this  last  proposition,  which 
is  true  for  absolute  numbers,  is  not  always  true  for  algebraic  ex- 
pressions ;  for  it  is  easily  shown  that  the  same  number  can  have 
more  than  one  square  root,  cube  root,  fourth  root,   (SfC. 

Let  us  denote  the  algebraic  value  of  the  square  root  of  a  by 
X,  and  the  arithinetical  value  of  it  hj  p;  we  have  the   equations 

oc^  =  a,    and    x"^  =  p"^,    whence    x  =^  ±  p. 

Hence  we  see,  that  the  square  of  p,  (which  is  the  root  of  «),  will 
give  a,  whether  its  sign  be   +   or   — . 

In  the  second  place,  let  x  be  the  algebraic  value  of  the  cube 
root  of  a,  and  p  the  numerical  value  of  this  root;  we  have  the 
equations 

x^  =  a,     and     x^  =  p^. 

The  last  equation   is  satisfied   by  making  x  =  p. 

Observing  that  the  equation  x'^  =/)•'  can  be  put  under  the  form 
^3  — p3  —  0,  and  that  the  expression  x^  — p'^  is  divisible  by  a:  — p 
(Art,  61),  which  gives  the  exact  quotient,  x'^  -{-  px  -{-  p^,  the  above 
equation  can  be  transformed  into 

(x  —  p)  (*^  +  poc  -{-  p"^)  =  0. 
Now,   every  value    of  x   which  will   satisfy  this    equation,  will 
satisfy  the   first  equation.     But   this    equation    can  be  satisfied    by 
supposing 

X — p  =  0,    whence    xz=p; 
or  by  supposing 

x^  -\-  px  -^  p"^  =  0, 
from  which  last,  we  have 

P         P     I — ir  /  —  1  ^  -y  —  3  \ 


CHAP.   VIII.]  CALCULUS    OF    RADICALS.  231 

Hence,    the    cube    root    of  a,    admits    of  three   different    algebraic 
values,  viz., 

P,     p[ r^ j.     and     p( -^ j. 

Again,  resolve  the  equation 

at*  =  p*, 

in  which  p  denotes  the  arithmetical  value  of  y  a .     This  equation 
can  be  put  under  the  form 

x^  —p^  —  0; 
which  reduces  to 

and  this  equation  can  be  satisfied,  by  supposing 

x^  —  ^-  =  0,     whence     x  :=.  ±.  p\ 
or  by  supposing 

3-2  _j_  p2  __  0,    whence    x  ^  — v  —  p^^  —  P  y  —  ^ 
We  therefore  obtain  four    different    algebraic  expressions  for  the 
fourth  root  of  a. 

As  another  example,  resolve  the  equation 

an6  _p6  _  0. 

This  equation  can  be  put  under  the  form 

(a;3  —  /)3)  (a;3  +  p3)  _  q  ; 

which  may  be  satisfied  by  making   either  of  the    factors  equal  to 
zero. 

But  j;3— p3  =  0,    gives 


,                 /_idrV-3\ 
X  =z  p,    and    X  =:  p  I ^ j. 

And  if  in  the  equation    x^  -{-  p^  =  0,   we  make  p  =  —  p',  it  be- 
comes   x^  —  p'^  =  0,  from  which  we  deduce 

.  =  ,',    and    ,=,.(:ii±./ZI), 

or,  substituting  for  p'  its  value  —  p, 

/-l±/^3\ 
X  =  —p,    and    x  =  —p  [^ -^ j. 

Therefore,  the  value  of  x  in  the  equation   x^  —  p^  =  0,  and  con 


232  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIII. 

sequently,  the  Gth  root  of  a,  admits   of  six  values.     If  we   make 


a  =z  -_ ,     and     a= 


2 
these   values  may  be  expressed  by 

p,    ap,    ap,    —p  —  ctp  —  a'p. 

We  may  then  conclude  from  analogy,  that  in  every  equation  of 
the  form  x^  —  a  =  0,  or  x"^  — p'"  =  0,  x  is  susceptible  of  m  dif- 
ferent values  ;  that  is,  the  m'^  root  of  a  number  admits  of  m 
different  algebraic  values. 

231.  If,  in  the  preceding  equations,  and  the  results  correspond- 
ing to  them,  vi'e  suppose,  as  a  particular  case,  a  =  1 ,  whence 
p  ■—  \,  we  shall  obtain  the  second,  tnird,  fourth,  &c.  roo'-s  of 
unity.  Thus  +  1  and  —  1  are  the  two  square  roots  of  unity, 
because  the  equation    a;-  —  1  =  0,  gives  a:  =  ±:  1. 

In  like  manner, 

-1  +  V-S       -1  ->/ —  3 

+  1,    -  ^ 


2  '  2  ' 

are  the  three  cube  roots  of  unity,  or  the  roots  of  x^  —  1  =  0 ;  and 


+  1,    -1,    +  V"^    -/^ 
are  the   four   fourth   roots   of  unity,  or  the   roots  of  a;*  —  1  =  0. 

232.  It  results  from  the  preceding  analysis,  that  the  rules  for 
the  calculus  of  radicals,  which  are  exact  when  applied  to  abso- 
lute numbers,  are  susceptible  of  some  modifications,  when  applied  to 
expressions  or  symbols  which  are  purely  algebraic ;  these  modifica- 
tions are  more  particularly  necessary  when  applied  to  imaginary 
expressions,  and  are  a  consequence  of  what  has  been  said  in 
Art.  230. 

For  example,  the  product  of 

y  —  a     by     y  —  a, 
by  the  rule  of  Art.  228,  would  be 

yf  —  a  X\/— a  =  \/+  a^. 

Now,  -yfcP-  is  equal  to  ±  a  (Art.  139);  there  is,  then,  ap- 
jKtrently,  an  uncertainty  as  to  the  sign  with  which  a  should  be 
all'ected.     Nevertheless,    the  true    answer    is    —  a\    for,  in  order 


CHAP.   VIII.] 


CALCULUS    OF    RADIC.A.LS. 


233 


to  square   \' m,   it  is   only  necessary  to  suppress  the  radical;   but 


V  —  aXy   —  a  =.  (-y/  —  0)2  =  —  a. 
Again,   let  it  be   required  to   form   the   product 

y/  —  a  X  \/  —  ^• 
By  ihe  rule  of  Art.  228,  we   shall  have 

^/  —  a  X  y/  —  b  =  y/ -\-ab. 

Now,  ^/ ah  =  zh  /J  (Art.  230),  p  being  the  arithmetical  vaiue, 
of  the  square  root  of  ah  ;  but  the  true  result  is  —  j)  or  —  -y/ ah, 
so  long  as  both  the  radicals  y  —  a  and  y  —  1^  are  a  fleeted 
with  the  sign   +• 


a 


1  ;    and    y  —  h 


For,     y  —  a  =  -y/  1 
hence, 

y/  —  a  X  s/  —  b  =  -J  a  .  yf  —  1  X  -y/      ~h  X  y 
=  Jah  X  —  1  =  —  J~d). 


By  similar  methods  we  find  the   different  powers  of 
be  as  follows  : — 


1    to 


1.       /  -  1  X  y  -  1  =  (/-  1)^  =  —  1. 

3.    (V^r  =  (/zTfr-.c/^T)^-  =  _  1  X  - 1  =  + 1. 

Again,  let  it  be  proposed  to  determine  the  product  of  \/  —  a 
by  the  y  —  b  which,  from  the  rule,  would  be  \/  +  ab,  and  con- 
sequently, would  give  the   four  values  (Art.  230), 

+  ya6,     —  y  ab,     -\- y  ah  .  -^ — 1,     —  y  ab  .  y —  1. 
To  determine  the  true  product,  observe  that 

V":rT=VT.V3T,   and  V^^=VT.V-T. 

ce  \J  -  a  .\/  —h=  \/ab.  y/  _  1 


But 


234  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIII. 

We  will  apply  llie  preceding  calculus  to  the  verification  of  the 
expression 

-1  +  7^^ 
2 
considered  as   a   root  of  the   equation    a;^  —  1  =  0 ;    that  is,  as  one 
of  the   cube  roots   of  1  (Art.  231). 
From  the  formula, 

{a  +  by  =za^  +  30%  +  Sab^  +b'\ 


we  have 


i^f^r 


'  -  (-l)^+3(-l)^V-3  +  3(-  1)  .  (7-  3)2  +  (V"^^ 


—  1  —  -v/  —  3 
The  second  value, ,    may  be  verified  in  the  same 

manner.  It  should  be  remarked,  that  either  of  the  imaginary  roots 
is  the  square  of  the  other  ;    a  fact  which  may  be  easily  verified. 

Theory  of  Exponents. 

233.  In  extracting  the  /i''''  root  of  a  quantity  a"",  we  have  seen 
that  when  ot  is  a  multiple  of  n,  we  should  divide  the  exponent  m 
by  n  the  index  of  the  root.  When  m  is  not  divisible  by  n,  the 
operation  of  extracting  the  root  is  indicated  by  indicating  the  di' 
vision  of  the  two  exponents.     Thus, 

m 

a  notation  founded  on  the  rule  for  the  exponents,  in  the  extrac- 
tion of  the  roots  of  monomials.  In  such  expressions,  the  numerator 
indicates  the  power  to  lohich  the  quantity  is  to  be  raised,  and  the 
denominator,  the  root  to   he  extracted. 

Q    t 2  .    i 7, 

Therefore,  y  d^  =  a^  ;     and     -y/  a'  z:^  a*. 

If  it  is  required  to  divide  a'"  by  o",  in  which  m  and  n  are  posi- 
tive whole  numbers,  we  know  that  the  exponent  of  the  divisor 
should  be  subtracted  from  the  exponent  of  the  dividend,  and  we 
have 


CHAP,   VIII.]  THEORV    OF    EXPONKNTS.  235 

If  m  >  n,  the  division  will  be  exact ;  but  when  m  <  n,  the  di- 
vision cannot  be  effected,  but  still  we  subtract,  in  the  algebraic 
sense,  the  exponent  of  the  divisor  from  that  of  the  dividend.  Let 
p  be  the  arithmetical  difference  between  n  and  m ;  then  will 

n  z:^  m  4-  p,     whence     p-  =  a  ?  : 

«""  1         1  1 

but  ■ — r-  —  — ;    hence,    a~P  =. — . 

a"'+P        aP  aP 

Therefore,  the  expression  a~P  is  the  symbol  of  a  division  which 
has  not  been  performed ;  and  its  true  value  is  the  quotient  repre- 
sented by  unity,  divided  by  a,  affected  with  the  exponent  p,  taken 
positively.     Thus, 

-3  ^  1  5  1 

a   •*=:—- ;     and     a~^  =  — -. 
a^  aP 

Since,     a~P  =  - —  ;     and     =  a?,    we  conclude  that, 

aP  a-P 

Any  factor  may  be  transferred  from  the  numerator  to  the  denom- 
inator, or  from  the  denominator  to  the  numerator,  by  changing  the 
sign  of  its  exponent. 

If  it  is  required  to  extract  the  «"»  root  of    — ,    we  have 

a"* 

,  n     /  ,  —m 

—  =:  a~"» ;     hence,      \/  —  =  V  a~^  =  a  "  ' 
a'"  ^   a"^ 

The  notation  of  fractional  exponents,  whether  positive  or  nega- 
tive, has  the  advantage  of  giving  an  entire  form  to  all  expres- 
sions whose  roots   or  powers  are  to  be  indicated. 

From  the  conventional  expressions  founded  on  the  preceding  rules, 
we  have 

wt  ,  n      /  1  in 

o"  =  \/a"';     a-P  =  — ;     and     V  —  =  a    "• 
aP  ^    a"* 

We  may  therefore  substitute  the  second  value  in  each  expres- 
sion, for  the  first,  or  reciprocally. 

As  aP  is  called  a  to  the  p  power,  when  p  is  a  positive  whole 
number,  so,  by  analogy, 

7n  _m 

a "  ,    a~P,    a    " , 

m 
are   called,  respectively,  a  to  the    —  power,  a  to  the   —  p  power 

71 


236  ELEMENTS    OF    ALGEBRA.  [CHAP     VIII. 

Ttl 

a  to   the power,  in  which   algebraists  have  generalized  the 

word  power.     It  would,   perhaps,  be   more  accurate  to  say,  a,  ex- 

m  ?n  .         , 

ponent    — ,    a,   exponent   — p,   and   a,  exponent    —  —  ;    using  the 

word  poiuer   only  when   we   wish   to   designate    the    product   of  a 
number  multiplied  by  itself  two  or    more  times. 

AlidtipUcation  of  Quantities  affected  with  any  Exponents. 

3  2 

234.  In  order  to  multiply  a^  by  a^,  it  is  only  necessary  to  add 
the  two  exponents,  and  we  have 


3  2  Sj,!  1 


—  -,  LA  

5     V    ^3    —   n^      '^    —   y/lS 


For,  by  Art.  233, 

a^  =  Y  a^  J     and    a^  ^=  y/  a^  ; 

hence,  a^  x  a^  =  y  o?  X  y  a"^ , 

reducing  to   a   common  index  (Art.  226),  and   then  multiplying, 

3  2  1,    . 19 

a^  X  a^  =    V  "'^  =  '^^^• 

_3.  5 

Again,  multiply  a   *     by    a^. 

_3  *    /Y  !i  g    . 

We  have,  a   *  —  V/  — r,    and     a^  =  ^/  a^  ; 

hence, 

a-^  Xai  =  \/l-X  ^^Jl/lx^V^oJ^/^^^lf,  ^  „A  , 
and  consequently, 

_3 


3  5  3,5  9    ,10  1_ 

-4  +  6    _  ^-12+12   =  „I2. 


In  general,  multiplying   a   "    by  a  ?  ;    we  have 


m    p 


a  "   X  ai  =  a  "     ?  =  a»</    . 

Therefore,  in  order  to  multiply  two  monomials  affected  with 
any  exponents  whatever, yoZ/oit;  the  rule  given  in  Art.  41,  for  quantities 
Hll'ected   with  entire  exponents. 


CHAP.   VIII.]  THEORY    OF    EXPONENTS.  237 

From  this  rule,  we  shall  find 

3_    _1  2.    3  U    A 2 

1.  a^b   ^c~i  X  aP'b^c^  z=  a*  b^c   ^. 

2.  3^-2*3  X  2a"5pc2  =  6a~  ^  b'^c^. 

-1  1  _J. 

3.  6a   U^c-""  X  5a^b-^c"  =  30a   ^^-i^n-m. 

Division  of  Qiiantities  affected  with  any   Exponents. 

235.  To  divide  one  monomial  by  another  when  both  are  affected 
with  any  exponent  whatever,  divide  the  co-efficient  of  the  divi- 
dend by  that  of  the  divisor,  for  a  new  co-efficient :  subtract  the 
exponent  of  each  letter  in  the  divisor  from  the  exponent  of  the  same 
letter  in  the  dividend,  and  then  annex  each  letter  with  its  new  ex- 
ponent. 

For,  the  exponent  of  each  letter  in  the  quotient  must  be  such, 
that,  added  to  the  exponent  of  the  same  letter  in  the  divisor,  the 
sum  shall  be  equal  to  the  exponent  of  the  letter  in  the  dividend  ; 
hence,  the  exponent  of  any  letter  in  the  quotient,  is  equal  to  the 
difference  between  the  exponent  of  that  letter  in  the  dividend  and 
in  the  divisor. 

EXAMPLES. 
2  3  2_^_3\  17 

1.  a^^a    4  =  a3    I    i)  ^a^i, 

2.  c 

3.  a'^  X  b^  -^  a~^b'^  =  aTo^»-8, 

4.  Divide       32a^b^c^    by    8aH''c~^.  Ans.  4a^bc*. 

5.  Divide       64a%^c~^    by    32a-%''^c~^.  Ans.  2a'^%\ 

Formation  of  Powers. 

236.  To  form  the  m"*  power  of  a  monomial,  affected  with  any 
exponents  whatever,  raise  the  co-efficient  to  the  m'^  poAver,  and 
for  the  exponents,  observe  the  rule  given  in  Art.  220,  viz,  multipli/ 
the  exponent  of  each  letter  by  the  exponent  m  of  the  power. 

For,  to  raise  a  quantity  to  the  m"^  power,  is  the  same  thing  as 
to  multiply  it  by  itself  m  —  1  times  ;  therefore,  by  the  rule  for 
multiplication,  the  exponent  of  each  letter  must  be  added  to  itself 
m  —  1   times,  or  multiplied  by  m. 


3  4  3_4  ]L 

,4      •     „5    _  ^'4     5    _  „     2  0 


238  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIII. 

/    3\5  15  /    2\3  6 

Thus,  \a*)=a^;    and     W)=a^=a^; 

I  1    3\6  9  /    _5\12 

also,  \2a   ^b^)  =  64a-3p  ;    and     \a   «)    =  a-io. 

1  *^ 

What  is  the  r/j"*  power  of   Sa^b'^c"^?  Ans.  S^'a^ b''^'"c'^"*. 

Extraction  of  Roots. 

237.  To  extract  the  n"^  root  of  a  monomial,  extract  the  «""  root 
of  the  co-efficient,  and  for  the  new  exponents,  follow  the  rule 
given  in  Art.  220,  viz.,  divide  the  exponent  of  each  letter  by  the 
index   of  the   root. 

For,  the  exponent  of  each  letter  in  the  result  should  be  such, 
that  when  multiplied  by  n,  the  index  of  the  root  to  be  extracted, 
the  product  will  be  the  exponent  with  which  the  letter  is  affected 
in  the  proposed  monomial ;  therefore,  the  exponents  in  the  result 
must  be  respectively  equal  to  the  quotients  arising  from  the  di- 
vision of  the  exponents  in  the  proposed  monomial,  by  n,  the  in 
dex  of  the  root.     Thus, 


also, 

The  rules  for  fractional  and  negative  exponents  have  been  easily 
deduced  from  the  rule  for  multiplication ;  but  we  may  give  a  direct 
demonstration  of  them,  by  going  back  to  the  origin  of  quantities 
affected  with  such  exponents. 

We  will  demonstrate  implicitly,  the  two  preceding  rules. 

711 

—  r 

Let  it  be  required  to  raise  a"    to  the power. 

s 

By  going  back  to  the  origin  of  these  notations,  we  find  that 


=  V  Y  J_  =  "'Ja-""-=  a""^;     hence. 


C")"'  = 


'      «     is  also     =  a    "*. 


CHAP.   VIII. 3  THEORY    OF    EXPONENTS.  239 

Rem.\rk  I. — The  advantage  derived  from  the  use  of  fractiona] 
exponents  consists  principally  in  this : — The  operations  performed 
upon  expressions  of  this  kind  require  no  other  rules  than  those 
established  for  the  calculus  of  quantities  affected  with  entire  expo- 
nents. This  calculus  is  thus  reduced  to  simple  operations  upon 
fractions,  with  which  we  are  already  familiar. 

EXAMPLES. 

2  a/~2  X  (3p 
1.  Reduce     — — ; — ^— ^    to  its  simplest  terms. 

V2 


1 


Ans.   4  y  3. 


2.   Reduce      < f  >     to  its  simplest  terms. 

1 2yT(3)2 ) 


Ans.   V  3 

384  ^ 


3.  Reduce     V   <  iz: f  >      to  its  simplest  terms. 


4.  What  is  the  product  of 

of  4.  0253  _{_  a^ii  ^  ab-^ah^  -{-  b^,  by    a 2  —  ^3, 

Ans.  a^  —  h^ 

5.  Divide        a^  —  a^b~'^  —  cflb  +  b^,    by    a'  —  b~^ . 

Ans.  cP"  —  h. 

Remakk  II. — 111  the  resolution  of  certain  questions,  we  shall  be 
led  to  consider  quantities  affected  with  incommensurahle  exj)onent&. 
Now,  these  incommensurable  exponents  may  be  treated  in  the 
same   manner    as    those    which    are    commensurable.      For,    let   us 

observe,  that  an  incommensurable  quantity,  such  as  v^  or  ^/\\., 
may  be  determined  approximatively,  as  near  as  we  please  :  hence 
we  can  always  conceive  it  to  be  replaced  by  an  exact  fraction, 
which  shall  differ  from  it  by  less  than  any  assignable  quantity. 
Having  done  this  we  can  apply  to  the  fraction  which  represents 
the  incommensurable,  the  rules  already  demonstrated. 


240  ELEMENTS    OF    ALGEBRA.  [CHAP.   VIU. 

Method  of  IndcteTminate  Co-efficients. 

238.  The  binomial  theorem  demonstrated  in  Art.  203,  explains 
the  method  of  developing  into  a  series  any  expression  of  the  form 
(a  +  5)"',  in  which  m  is   a  whole  and  positive   number. 

Algebraists  have  invented  another  method  of  developing  alo-e- 
braic  expressions  into  series,  called  the  method  by  indeterminule 
co-efficients.  This  method  is  more  extensive  in  its  applications, 
can  be  applied  to  algebraic  expressions  of  any  nature  whatever, 
and  indeed,  the  general  case  of  the  binomial  theorem  may  be  de- 
monstrated by  it. 

Before  considering  this  method,  it  will  be  necessary  to  explain 
what  is  meant  by  the  term  function. 

Let  a  ^  b  -\-  c. 

In  this  equation,  a,  b,  and  c,  mutually  depend  on  each  other  for 
their  values.     For, 

a  :=  b  -\-  c,     b  =  a  —  c,     and     c  =  a  —  b. 

The  quantity  a  is  said  to  be  a  fu7iction  of  b  and  c,  b  a  function 
of  a  and  c,  and  c  a  function  of  a  and  b.  And  generally,  when 
one  quantity  depends  for  its  value  on  one  or  more  quantities,  it  is 
said  to  be  function  of  each  and  all  the  quantities  on  which  it  de- 
pends. 

239.  If  we  have  an  equation  of  the  form, 

A-\-  Bx  -^  Cx^  +  Dx^  +  Ex^  +  &c.  =  0 ; 

it  is  required  to  find  the  values  of  the  co-efficients  A,  B,  C,  D, 
E,  &c.,  under  the  following  suppositions  : 

1st.  That  no  one   of  the   co-efficients   is   a  function  of  x. 

2d.  That  the  series  shall  be  equal  to  zero,  whatever  be  the 
number  of  its  terms ;    and 

3d.  That  it  shall  be  equal  to  zero,  whatever  value  may  be  at- 
tributed to  X. 

Now,  since  the  co-efficients  are  independent  of  x,  their  values 
cannot  be  affected  by  any  supposition  made  on  the  value  of  .r : 
hence,  if  they  be  determined  tor  one  value  of  x,  they  will  be 
known  for  all  values  whatever. 


CHAP.  VIII.]  INDETERMINATE    CO-EFFICIENTS.  '241 

Let  us  now  make 

T  =  0,     which  gives 
Bx  +  Cx2  +  Dx^  +  Ex*  +  &c.  =  0 ; 
and  consequently,  Jl  =  0. 

If  we  divide  by  x,  we  have 

B  +  Cx  +  Dx"^  +  Ex^  +  &c.  =  0  , 
and  by  again  making  a:  =  0,  we  have 

Cx  +  Dx2  +  Ex^  +  &c.  =  0 ; 
and  consequently,  B  =  0 

Dividing  again  by  x,  we  have 

C  +  Da:  4-  iia;2  -f  &c.  =  0; 
and  by  again  making  a;  =  0,  we  obtain 

Dx  4-  Ex'i  +  &c.  =  0, 
and  consequently,  C  ^  0  ; 

and  by  continuing  the  process  we  may  prove  that,  each  co-efficient 
must  be   separately  equal   to   zero. 

It  should  be  observed,  that  A  may  be  considered  the  co-efficient 
of  x^. 

240.  The  principle  demonstrated  above,  may  be  enunciated  un- 
der another  form.     If  we  have   an    equation   of  the   form 

a -\- hx -{-  cx^  -{-  dx"^  +    .   .   .    =  a'-\-  h'x  +  c'x"^  +  d'x:^  +    .   .   . 
which    is  satisfied  for  any  value  whatever  attributed  to  x,  the   co- 
efficients   of  the   terms  involving   the   same  powers   of  x   in   the   two 
members,  are  respectively  equal.     For,  by  transposing  all  the  term.? 
into  the  first  member,  the  equation  will  take  the  form 
A-\-Bx+  Cx-^  +  Dx^  -\-  Ex*  +  &c.  =  0  ; 
whence,       a  —  a^=z  0,     b  —  b'=  0,     c  —  </=  0  .... ; 
and  consequently, 

a  =  a',     b  =  y,     c  =  c\     d  =  d^  .  .  .  . 

Every  equation  in  which  the  terms  aie  diranged  with  reference 
to  a  certain  letter,  and  which  is  satisfied  for  any  value  which 
may  be  attributed  to  that  letter,  is  called  an  identical  equation,  in 
order  to  distinguish  it  from  a  common  equation,  that  is,  an  equa 
tion  which  can  only  be  satisfied  by  particular  values  of  the  un- 
known quantity. 

16 


242  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIII. 

241.  Let  us  apply  the  above  principles  in  developing  into  a 
series  the  expression 

a 
a'  +  b'x' 

It  is  plain,  that  any  expression  equal  to  the  above,  must  con- 
tain X,  or  quantities  depending  on  x ;  and  a,  a,  b' ,  or  quantities 
depending  on  them.     Let  us  then  assume 

,    ",,    =  A-\-  Bx+  Cx^  +  Dx^  +  Ex'^  +  &c.  .  .  .  (1), 
a^  -{-  b  X 

in  which  the  co-efBcients  A,  B,  C,  D,  &c.,  are  functions  of  a,  a', 
//,  and  independent  of  x.  These  are  called,  indeterminate  co-effi- 
cients. It  is  required  to  find  their  values  in  terms  of  a,  a',  b\  on 
which  they  depend. 

For  this  purpose,  multiply  both  members  of  equation  (1)  by 
a^  +  ^'*-  Arranging  the  result  with  reference  to  the  powers  of 
X,  and  transposing  a,  we  have 


,    Aa'  +  Ba' 
''=^  +Ai' 


i^: 


X  4-  Ca' 
+  Bb' 


a?2  +  Da'  I  a;3  +  Ea' 
+  Cb'l      -f  Db' 


x'-h  .  •  .  (2), 


and  since  this   equation   is   satisfied  for  any  value  of  x,  we  have 
(Art.  239), 

a 

Aa'  —  a  =1  0,     whence,     A  =  -—; 

a 

also,  Ba'  -\-  A¥  =  0,     Avhence, 

Ay  _  _a_      ^_  __ob' 
a'  a'       a'  a'"'' 

also,  Ca'  +  Bb'  =  0,     whence. 


Bb'  /       ab'^ 


b'        ab'^ 

X  —  = 
a  a' 


r,r3 


also  Da'  +  Cb'  =  0,     whence, 

a/_ab2  b'  ab'3 

~         a'   ~   a/-^  a'  ~         a'-*' 

It  is   plain   that  the  terms   will  be   alternately  plus   and  minus, 

and  that  the  co-efficient  of  any  term  is  formed  by  multiplying  that 

b' 
of  the  preceding  term  by    —  —7  5    therefore,  we  have 

-        ab'       ,    ab'^    ,       ab'3    ^    ,    ab"^    ,        . 


</  -f  b'x        a'        a'2  a- 


CHAP.   VIII.] 


INDETERMINATE    CO-EFFICIENTS. 


243 


242.  The  method  of  indeterminate  co-ejficients  requires  that  we 
should  know  the  form  of  the  development.  The  terms  of  the  de- 
velopment are  generally  arranged  according  to  the  ascending  powers 
of  .r,  commencing  with  the  power  a?" ;  sometimes,  however,  this 
form  is  not  applicable,  in  which  case,  the  calculus  detects  the 
error  in  the  supposition. 

For  example,  develop  the  expression -. 

3a;  —  x^ 

Let  us  suppose  that 

— ^-—  =A  +  Bx-\-Cx^-\-Dx^-\-  .  .  .  ., 
3x  —  x^ 

whence,  by  reducing  to  entire  terms,  and  arranging  with  reference 
to  X, 

0  =:  —  1  +  2Ax  +  35    a;2  +  3C    a;3  +  3i)    a;*  +   .  .  .  ., 
-A  -B  - C 

whence  (Art.  239), 

—  1=0,     3A=0,     3B  —  A  =  0  .  .  .  . 
Now,  the  first  equation,    —1=0,   is  absurd,  and  indicates  that 

the  above  form  will  not  develop  the  expression — .     But  if 

3x  —  x'^ 

we   put  the   expression  under  the  form    —  x  — ,    and  make 

a;        3  —  X 


3  —  X 


=  A  -{-  Bx  -\-  Cx^  -\-  Dx^  i- 


we  shall  have,  after  the  reductions  are  made. 


C3A  +  3B 
~   l-l-A 

x-f  3C 
-B 

a:2-f  3D 
-  C 

x^  + 

which  gives  the  equations 

3.4  -  1  =  0,     3B 

~A  = 

[),     3C  - 

B  =  0 

whence.       A  = 


^  =  1'  ^  =  .^-  ^  =  ^- 


Therefore, 

1  1/11  1  1 

3  9      ^27  81       ^ 


■). 


tliat  is,  the  development   contains  a   term  with   a   negative   expo 
nciU 


244  ELEMENTS    OF    ALGEBRA.  ICHAP.  VIII. 

Recurring  Series. 

243.  The  development  of  algebraic  fractions  by  the  method  of 
indeterminate  co-efficients,  gives  rise  to  certain  series,  called  re- 
curring series. 

A  recurring  series  is  the  development  of  a  rational  fraction  in- 
volving X,  made  according  to  a  fixed  law,  and  containing  the  ascend- 
tng  powers  of  x  in  its  different  terms. 

It  has  been  shown  in  Art.  241,  that  the  expression 
a  a         aV  ah''^     ^        ab'^     ^ 

in  which  each  term  is  formed  by  multiplying  that  which  precedes 

It  by X. 

a 

This  property  of  determining  one  term  of  the  development  from 
those  which  precede,  is  not  peculiar  to  the  proposed  fraction ;  it 
belongs  to  all  rational  algebraic  fractions,  and  may  be  thus  ex- 
pressed ;  viz..  Every  rational  fraction  involving  x,  may  he  developed 
into  a  series  of  terms,  each  of  lohich  is  equal  to  the  algebraic  sum 
of  the  products  which  arise  from  multiplying  certain  terms  of  a  par- 
ticular expression,  by  certain  of  the  preceding  terms  of  the  series. 

The  particular  expression,  from  which  any  term  of  the  series 
may  be  found,  when  the  preceding  terms  are  known,  is  called  the 
scale  of  the  series  ;  and  that  from  which  the  co-efficient  may  be 
formed,  the  scale  of  the   co-efficients. 

b' 

In  the  preceding  series,  the  scale  is -x,  and  the  series  is 

a 

b' 
called  a  recurring  series  of  the  first  order ;  and -is  the  scale 

of  the  co-efficients. 

T        ■    ,  IT,  a  -^-hx 

Let  It  be  required  to  develop ; — r    into  a  series. 


Assume 

a  -\-  bx 


A  +  Bx+Cx-^^  Dx^  +  Ex*  -f   . 


reducing  to  entire   terms,   and  transposing,  we  have 


X  -(-  Ca'  I  x^  +  Da^ 
+  Bb'  I  +  Cb' 
+  Ac'l      +  Be" 


x^  -f  Ea' 
+  Db' 
+  Cc" 


X*  + 


CHAP.  Vin.]  INDETERMINATE    CO-EFFICIENTS.  245 

which  gives  the  equations 

a 
Aa'  —  a=  0,     or     A  =  —-, 

b'  b 

Ba'  +  Ah'  —   b   =0,     whence,     B  — tI  -f-  — , 

a  a 

y  (/ 

Caf  +  Bb'  -{-A</=0,     whence,     C= -B -A, 

of  a 

V  c' 

Daf  +  Cb'  +  5</  =  0,     whence,     D= -C -B, 

a  a 

h'  cf 

Ea'  +  BV  +  Ct/  =  0,     whence,     E- -D -C. 

a  a 


From  which  we  see,  that  the  first  two  co-efficients  are  not  ob- 
tained by  any  law ;  but  commencing  at  the  third,  each  co-efficient 
is  formed  by  multiplying  the   two   which  precede  it,  respectively, 

h'  cf 

by 7    and -,    viz.,  that  which  immediately  precedes  the 

a  a 

b' 

required    co-efficient  by ,    that  which  precedes  it  two  terms 

a' 

c' 
by -,    and  taking  the  algebraic  sum  of  the  products.     Hence, 


/_  b'        c\ 


is  the  scale  of  the  co-fiflcients. 

From  this   law  of  the   formation  of  the  co-efficients,  it  follows 
that  the  third  term  of  the  series,   Cx^,  is  equal  to 


-^Bx^-^Ax^; 
a  a 

-X  .  B  X -x^  .A 

a  a 


In  like  manner,  we  have  for  Dx^^ 

h'  cf 

_  iL  Ca;3 Bx^  ; 


X  .  Cx"^ a;2  .  Bx. 

a'  a 


Hence,  each  term  of   the    required   series,   commencing   at   the 
third,  is  obtained   by  multiplying    the    two    terms    which   precede 


246  ELEMENTS    OF    ALGEBRA.  [CHAP.   VIII 

respectively  by 

and  taking  the   sum  of  the  products  :    hence,  this   last   expression 
is  the   scale  of  the  series. 

Recurring  series  are  divided  into  orders,  and  the  order  is  esti 
mated  by  the  number  of  terms  in  the  scale  which  involve  x. 

Thus,  the  expression   — ~   gives   a  recurring    series   of  the 

h' 

first  order,  the  scale  of  which  is x. 

a 

The  expression   — -,- — -    gives  a  recurring  series  of  the 

^  a'  +  b'x  H-  c'x^    "  ^ 

second  order,  of  which  the  scale  is 

\       a'  a'      I 

The  series  obtained  in  the  preceding  Art.  is  of  the  second 
order. 

In  general,  an  expression  of  the  form 

a  -\-   hx  -\-  cx"^   +    •   •   •  kx^—'^ 
a'-\-  b'x  +  </x2  +    ...  A'a;» 
gives   a  recurring  series   of  the  n^^  order,  the  scale  of  which  is 

I 7  «> a;2  .   .   . 7  a;" ) . 

\       a  a  a        / 

Remark. — It  is  here  supposed  that  the  degree  of  x  in  the  nu- 
merator is  less  than  it  is  in  the  denominator.  If  it  was  not,  it 
would  first  be  necessary  to  perform  the  division,  arranging  the 
quantities  with  reference  to  x,  which  would  give  an  entire  quo- 
tient, plus  a  fraction  similar  to  the  above. 

r^,         .     ,,  .        1  -x-3x^  +  4x^  +  x*        . 

Ihus,  m  the  expression    —cT^ 5 — .    gives 

2  —  5a:  -L  3x^  —  x'^ 


X*  +  4a!3  —  3a;2  —  a;  +  1 
-f  7a;3  —  8x2  +  0; 


—  a;3  +  33-2  —  5ar  +  2 

-a;   -~7  ■ 


+  13a;2  —  34a;  +  15. 
Performing  the    division,  we   find   the  quotient  to  be    —  x  —  7. 
plus  the  fraction 

13x2  —  34a?  +  15      ^    15  -  34a;  +  13a;2 
-  -  a;3  +  3x2  —  5a:  +  2  "  2  —  5a:  +  3x2  —  ^^3 


CHAP.  VIII. 1  BINOMIAL    THEOREM.  247 

Demonstration  of  the  Binomial   Theorem  for  any  Exponent. 

244.  It  has  been  shown  (Art.  61),  that  any  expression  of  the 
form 

^m  _  ym^    ig  exactly  divisible  by   a;  —  y, 

when  77J  is  a  positive  whole  number.     That  is, 

npTtl    fl#ffl 

X  —  y  J  p  y  3 

The  number  of  terras  in  the  second  member  is  equal  to  m ;  and 
if  we  suppose  x  =  y,  each  term  will  beco.me  equal  to  x^—^ : 
hence, 

X"'  —  x" 


,^771  ^m 

=  mx" 


X  —  X 

We  propose  to  prove  that  the  quotient  will  have  the  same  form 
when  m  is  negative,  and  also,  when  ?n  is  a  positive  or  negative 
fraction. 

First,  when  m  is  a  whole  number,  and  negative. 
Let  71  be  a  positive  whole  number,  and  numerically  equal  to  m 
Then, 

?ra  =  —  n. 
By  observing  that 

—  x~"y~''^  X  (a;"  —  y")  =  x~"  —  y""", 
we  have 

—  =  —  x-'^y-"  X  ^^ ^—  =  —  a;-2"«a:'»-^  =  —  7«x~"~\ 

X  —  y  X  —  y 

after  making  y  =  x  ;    and  by  restoring  m, 


—  7*,^m    1 


mx"^  \    77?  being  a  negative  number. 

Second,  let  m  he  a  positive  fraction,  or    m  =z  — . 

? 
1  E 

Let  x^  =  V,     whence,     xi  =.vp,     and     a:  =  u?  ; 

i  E 

and  yi  ^=:u,     whence,     y<i  =  uP,     and     y  z=.  ui . 

p          p  yP  —  nP 

TT                          **  —  V'        i^''  —  w?           V  —  u 
Hence,  ^—  = = 

X  —  y  i;2  —  w3  tJ9  —  M?  ' 


248  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIII 

If  Ave  suppose    x  ■=  y,  we  have  v  =  w  ;    and  since  p  and  q  are 
positive  whole  numbers,  we  have 
p         p 

X  —  a;  gvi  q  q 

fter  substituting  for  v  its  value    x^. 
If  we  restore  m,  we  have 

^771  yin 

==  mx'^~'^,    m  being  a  positive  fraction. 

X  —  X 

Third,  let  m  be  a  negative  fraction,  or    m  z=z —, 

1 
i  -2. 

Let  xi  z=z  V,     whence,     a:    «  =  v~p,     and     a:  =  v? ; 

i  -t 

and  yi-=iu,     whence,     y  1z=iu~p,     and     yz=ziii. 

p  p  v-P  —  n-P 


X  g  —  y  q       1)  p  —  u~P  V  —  u 

Hence, —  = = 

X  —  y  v'i  —  ui  v'l  —  ui 


V  —  u 

But  since  —  jo  is  a  negative  integer,  and  q  a  positive  integer, 
we  have  from  what  has  preceded,  after  making  ar  =  y,  which 
wives    U  :=z  V, 


X  g  —  X  i        —  pv~P-^ 


J—  v~P~i ; 


X  —  X  qVi    '  q 

J. 

and  substituting^ for  v  its  value    x^    we  have 

_v_         _p 

X  1  —  X  1  v  p    ---1 

=  —i-t)-p-q  — £-.x   1       ; 

a;  —  X  q  q 

and  restoring  in,  we  have 

f^m    rvytn 

=  mx'"~\    m  beinsr  a  negative  fraction. 


245.  We  are  now  prepared  to  find  a  general  formula  for  the 
development  of  the  binomial  [a  +  Z/)™,  in  which  the  exponent  m  is 
positive   or  negative,  and  either  integral  or  fractional. 

In    order    to    simplify    the    process,    let    us    place    the    binomial 


CHAP.   VIII.]  BINOMIAL    THEOREM.  249 

under  the  form 

(^  \  /  fix"" 

1  +  —  j  ]-  =  a-  (l  +  —  j     (An.  220). 

/           b  \"* 
If  we  find  the   development  of    |1  -\ j   ,    and    then   multiply 

It  by  a"",  the  product  will  be  the  development  of  (a  +  />)"*. 
In   order  further  to  simplify  the  expression,  let  us  make 

b 

—  =zx; 
a 

then,  the  binomial  to  be  developed  will  be  of  the  form 

(1  4-  a-)-. 
As  this  development  must  be  expressed  in  terms  of  ar,  and  kno\vn 
quantities  dependent  for  their  values  on  1  and  m,  we  may  assume 

(1  +  x)"*  =  A  4-  Bx  +  Cx2  +  Z)x3  +  Ex^  +  &c.   .   .   .  (1), 

in    which   the    co-efficients    A,  B,   C,  Sic,  are   independent  of  x, 
and  functions  of  1   and  m. 

Now,  since  this  equation  is  true  for  any  value  of  x,  if  we  make 
X  =z  0,  we  have 

(1)'"  =  ^  =  1. 
Substituting  this   value   in  equation   (1),   we  have 
(1  +  a-)-"  =  1  +  J5x  +  ar2  +  Dx^  +  Ex^  +  &c.   .   .   .   (2). 

Since  the  form  of  the   above  development  will  not  be  changed 
by  placing  y  for  x,  we  may  write 

(1  +  y)'"  =  1  +  5y  -(-  Cf  +  Df  4-  E?/  4-  &c.   .   .   .  (3). 

Subtracting  equation  (3)   from  (2),   and  dividing   both  member 
by   X  —  y,  we  have 

('+')''-i'±yT^B^'^^^+rJ-^^^D^^^^^h&c.  . .  .(4) 
X — y  X — y  X  —  y  x — y 

Make     I  ■}-  x  =  v,   and    1  4-  y  =  «  ;    whence,    x  —  ij  ^:  v  —  u. 
Substituting   these   values   in  the   first  member  of  equation  (4) 
and  we  have 

V-  -n-  ^  ^(x-y)  ^  C^-^-^^  +  D^^^^  +  &o.  ..  .(5) 
V  —  u  X  —  y  x  —  y  x  —  y 

If  now,  we  make 

X  z=  y,    whence,    v  =  v. 


250  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIII 


we  have,  from  Art.  244, 

v"»  —  u"*        v'"  —  d" 


-1  =  ot(1  4-  a:)"»-i  ; 


V   —  U  V  —  V 

while  the  quotients  in  the  second  member  become,  respectivel)', 


2 

1,      ".: ^  =  2a;2-i  =  2x; 


X  —  y  x'^  —  y         „  „   , 

J    1  J       0-v,2— 1 


X  —  y  X  —  y 

'^    ~y    =  3a;3-i  =  3x"  ;      ^    ~^    =  4x*-i  =  4^;^  ;    &c. 
X  —  y  ^  —  y 

Substituting  these  values  in   equation   (5),  and  we  have 

m{l+  xy-^    =    +  B  +  2Cx  +  3Dx^  +  4Ex^  +  &c (6). 

Multiplying  both   members  of  this    equation  by    I  -\-  x,  and  ar- 
ranging the  second  member  with  reference  to  x,  and  we  have 


'0^"t5 


m(l  +x)'"  =  5  +  2C 
+    B 


X  -\-  SDlx"^  +  AE 
+  2C I       +  3D 


x'^  +  &c 


If  we  now  multiply  equation   (2)  by  m,  Ave   have 

jfi  n  j^  x'^m  z=  m  -\-  mBx  +  mCx^  +  mDx^  +  mEx^  +  &c. 

If  we  place  the  second  member  of  the  last  two  equations  equal 

to  each  other,  we  shall  obtain  an  identical  equation.     Then,  placing 

the    co-efficients    of   the    like    powers    of  x    equal   to    each    other 

(Art.  240),  we  have 

m 
B  =  m,      whence,  B  =z  —  ; 

^     B(m  —  1)      m(m  —  1) 
2C+   B=mB,   whence,    C=     ^  '—     ^  '• 


3D+2C=mC,   whence,   D 


2  1.2' 

C(m— 2)      m{m—l)(m—2) 


4E-\-3D=TnD,   whence,    E  = 


3  1.2.3 

D(m— 3)      m{m—l)(?n—2){m~3) 


4  1.2.3.4 

&c.,  &c. 

Substituting  these  values  of  A,  B,  C,  D,  &c.,  in  equation  (2), 
we  obtain 

m(m — 1)    „       m  (m  —  I)  (m  —  2)    , 
{l+xr=.l+mx+  ^\    ^    '^^  +  1  \    a     '^      3—^  ^' 

m{m-l)(m-2){m-3)  ^^       ^^ 
^1.2.3.4  ^ 


CilAP.    VIII. 1  BINOMIAL    THEOREM.  251 

If  we  now  replace  x  bv  its  value    — ,    we  have 

a 

/  h  \"^  b       m{m—\)b'^      m(m —  I)  (m  —  2)  b^ 

(l+-)    =l  +  m--\-  ^      ^     ^  +  -,  o  o       -+&C. 

\  a  /  a         1.2        a-^       1.2.      3         o'^ 

Finally,  multiplying  by  a"*,  we  obtain 

m  (m  —  1)        ,,. 

(a  -f  b)'"  =  a"*  +  ma^-^b  -\ ^^ -'  O^-^b^ 

1  *         A^ 


7n  im  —  ^)  (fn  —  2) 
+  ^i        2  3        ""  ^  ' 

a  development  which  is  of  the  same  form  as  the  one  obtained  in 
Art.  203,  under  the  supposition  of  m  being  a  positive  and  whole 
number. 

Ajii^Ucations  of  tlie  B'moinial   Theorem. 

If  in  the  formula  (a;  +  «)"*  = 

(a  m  —  \     a?  m  —  \    m  —  2a^  \ 

1  +OT.  —  +  m.—-—.—  +  w2.—-— .-——.—  +  .  .  . 
a;  2         a"'  2  3  a;'^  / 


1       .  -  n  / 

we  make    m  =  ■ — ■,    it  becomes    (,r  +  «) "    or    V  a;  +  a  = 
n 

-i-l  1-1    i-3 

J(,+l.Ji  +  l.iL^.4  +  l    «    _.iL.       «V  .  ,  .) 

\         n      X        n  2         x^       n  2  Z         :r  I 

or,  reducing,  \J  x  -^  a  — 

i/,    ,1       a         1     K— 1     a2        1     n_l    2re—  I     a3  \ 

>r"    IH -. . . 1 . . . ..     ..) 

\  11      X        n       2n       x^       n       2n  3«         x^  I 

The    fifth   term    can    be     found   by    multiplying    the    fourth    by 

— — —   and  by   — ,    then    changing  the   sign  of  the   result,  and 

so  on. 

Remark. — If  in  this  formula,  we  make  n  =  2,  n  =  3,  n  =  4, 
&c.,  the  development  will  become  the  approximate  square  root, 
cube  root,  fourth  root,  &c.,  of  the  binomial  {x -\-  a)  \  and  by  as- 
signing values  at  pleasure  to  x  and  a  as  well  as  to  ti,  we  can 
find  any  root  whatever  of  any  binomial.  If  m  is  negative,  or  frac- 
tional, there  will  be  no  limit  to  the  number  of  terms  to  which  the 
series  may  be  carried.     Such  a  series  is  called  an  infinite  series. 


252  ELEMENTS    OF    ALGEBRA  [CHAP.  VIII, 

The  binomial  formula  also  serves  to   develop  algebraic  expres- 
sions into  series. 

Take,  for  example,  the  expression ,    we  have 

1  —  z 

_L_  =  (1  _  2)-i. 

In  the  binomial  formula,  make   ?7i  =  — 1,    x:=z\,    and   a  =  —  Zi 
it  becomes 

(1  _  ^)-i  =  1  _  1  .  (_  ;r)  -  1  .  ^—  .  (-  :^? 
-1-1      -1-2       .         -3 

or,  performing   the    operations,   and    observing   that   each    term   is 
composed  of  an  even  number  of  factors  affected  with  the  sign  — 

(1  —  Z)-^   = =   1  +  2  +  ^2  +  ^3  +  S*  +  05  +     .    .    .    . 

The  same  result  will  be  obtained  by  applying  the  rules  for  di- 
vision (Art.  55). 


1 

1st  remair 

ider 

-      +:^ 

2d    -      - 

- 

-      +z^ 

3d    -      - 

- 

-        +23 

4th  -      - 

- 

-        +   X* 
+     . 

1  -]- z  +  z^  +  z^ -\- z*  + 


Again,  take  the  expression    — —,    or    2(1—2)" 


We  have  2(1  —  z)-^  = 

2[l-3.{-z)-3.^^.{-zY-3.=^.--f^.{-^y-.]; 
or         2  (1  -  2)-3  ^  2  (1  +  32  +  6^2  +  iQz^  +  152*  +   .   .  .) 


To  develop  the  expression    y  22  —  2^,  which  reduces  to 
Vi^f]  —  ^)^    we  first  find 


=  1 2 2"^ 2 

6  36  618 


CHAP.   VIII.]  BlNOiMlAL     THEORKM. 

hence 


253 


V2._.»=vs(i-|^-^^'-^-'-.H 


EXAMPLES. 

1.  To    find   the   value  of    r — -— -- =  (a  +  &)-2    in    an    infinite 

(a  +  by 

iries. 

2.  To  find  the  value  of in  an  infinite  series. 

r  -\-  X 

x^       x^        x'^ 
Ans.  r  —  oc  -\ +  -^,  &c. 

3.  Required  the  square  root  of    -^ in  an  infinite  series 

cc^         x' 


4.  Required  the  cube  root  of    — -„ -—    in  an  infinite  series. 

40x6 


1  /  2x2  5^4  40x6 


Remark. — AVhen  the  terms  of  a  series  go  on  decreasing  in 
value,  the  series  is  called  a  decreasing  series  ;  and  when  they  go 
on  increasing  in  value,  it  is  called  an  increasing  series. 

A  converging  series  is  one  in  which  the  greater  the  number  of 
terms  taken,  the  nearer  will  their  sum  approximate  to  the  true 
value  of  the  entire  series.  When  the  terms  of  a  decreasing  and 
converging  series  are  alternately  positive  and  negative,  we  can,  by 
taking  a  ^iven  number,  determine  the  degree  of  approximation. 

For,  let  a  —  h  -\-  c  —  d  -\-  e  —  f  -\-  .  .  .,  &c.,  be  a  decreasing 
series,  b,  c,  d,  .  .  .  being  positive  quantities,  and  let  x  denote  the 
true  value  of  this  series.  Then,  if  n  denote  any  number  of  terms, 
the  value  of  x  will  be  found  between  the  sum  of  the  n  and  n  +  1 
terms. 

For,  take  any  two  consecutive  sums, 

a  —  b  -{-  c  —  d  -\-  e  — f,    and    a  —  b  -{-  c  —  d  -{-  e  — f-{-  g 
In  the  first,  the  terms  which  follow  — /",  are 


254 


ELEMENTS    OF    ALGEBRA. 


ICHAP.  VIII. 


but  since  the  series  is  decreasing,  the  differences  of  the  consecu- 
tive terms  g  —  h,  k  —  I,  .  .  .  are  positive  numbers ;  therefore,  in 
order  to  obtain  the  complete  vaUie  of  a?,  a  positive  number  must 
be  added  to  the  sum    a  —  b-\-c  —  d-{-e—f.     Hence,  we  have 

a  —  b  -\-  c  —  J  +  e  — y<  X. 

In  the  second  series,  the  terms  which  follow  +  g,  are  —  h-{-  k, 
—  I  -^  m  .  .  .  .  Now,  the  differences  —  h  -\-  k,  —  I  -{-  m  .  .  ., 
of  the  consecutive  terms,  are  negative ;  therefore,  in  order  to  ob 
tain  the  sum  of  the  series,  a  negative  quantity  must  be  added  to 

a  —  b-{-c  —  d  +  e  — /+  g, 

or,  in  other  words,  it  is  necessary  to  diminish  it.     Consequently, 

a  ^  b  -\-  c  —  d  -\-  e  — f  -j-  g  "^  x. 

Therefore,  x  is  comprehended  between  the  sum  of  the  n  and  n  +  1 
terms. 

But  the  difference  between  these  two  sums  is  equal  to  g;  and 
since  x  is  comprised  between  them,  their  difference  g  must  be 
greater  than  the  difference  betv.'een  a:  and  either  of  them ;  hence, 
the  error  committed  hrj  takmg  n  terms,  a  —  b  +  c  —  d  +  e  —  f,  of 
the  series,  for  the  value  of  x,  is  numerically  less  than  the  follow- 
ing term. 

Summation  of  Scries. 

246.  An  interesting,  and  at  the  same  time  useful  application  of 
the  principles  involved  in  the  summation  of  series,  is  found  in 
determining  the  number  of  balls  or  shells  contained  in  a  given  pile, 

Let  ABC  be  a  triangular  pile  of  balls, 
having  eight  balls  on  each  of  the  three 
equal  lines,  AB,  BD,  and  AD,  and  also, 
eight  balls  in  height  along  the  line   CB. 

Now,  the  proposed  pile  consists  of  8 
horizontal  courses,  and  the  number  of 
shot  in  each  course,  is  the  sum  of  an 
arithmetical  series  of  which  the  first  term 
is  1,  the  last  term  the  number  of  courses 
from  C,  and  the  number  of  terms,  also 
the  number  of  courses   from  C.     There- 


CHAP.    VIII.] 

STIMMATIOX    OF    SERIES. 

•^55 

fore  we  have 

1st 

course  is  equal  to    (1  +  1)  x 

1=    1 

2(1 

(1  +  2)  X 

1     =    3 

Sd 

(I   +   3)    X 

11=    6 

4tli 

(1   +  4)    X 

2     =  10 

5th 

(1  +  5)  X 

21  =  15 

6th 

(1  +  6)  X 

3    =21 

7th 

(1  +  7)  X 

31  =28 

8th 

(1  +  8)  X 

4    =  36. 

Hence,  the  num 

3er  of  shot  in  the  pile  will  be  equal  to  the  sura 

of  the  series 

1,    3,    6,    10,    15,    21,    23,   36; 

in  which  any  term  is  found  by  adding  1  to  the  number  of  the  term 

and  multiplying  the  sum  by  half  the  number  of  terms. 

Thus,  if  we  suppose  the  horizontal  layers  to  be  continued  down, 

and  denote  the  number  of  any  layer  from  the  top  by  n,  we  shall 

have 

nin  +  1) 
1,    3,   6,    10,    15,    21,  ...   .  -^^-; 

and  the  sum  of  this  series  Avill  express  the  number  of  balls  in  a 
triangular  pile,  of  which  n  denotes  the  number  in  either  of  the 
bottom  rows. 

If  the  general  term  of  any  increasing  series  of  numbers  involves 
n  to  the  m'^  degree,  the  sum  of  the  series  will  not  involve  n  to 
a  higher  degree  than  (m  +  1).  For,  the  sum  of  such  series  can- 
not exceed  n  times  the  general  term,  and  hence,  cannot  involve 
n  to  a  higher  degree  than  m  +  1.     Let  us  therefore  assume 

n{n-\r\) 


1  +  3  +  6+10  +  15 


=  A  +  Bn+Cn'^  +  Dn\ 


in  which  the  co-efficients  A,  B,  C,  and  D,  are  not  functions  of  n. 
In  order  that  these  co-efficients  may  be  determined,  we  must  find 
four  independent  equations  involving  them.     If  we  make 

n=l,     we  have     A  +  B  +  C  +  D  =  1  =1  (1), 

«  =2,  gives  A  + 2B  +    4C+    8Z)=l+3  =4  (2), 

n  =  3,      "       A+3B+    90  +  277)=  1 +3  +  6  =10  (3), 

n=4,      "      A  +  4^+  16C+  64D=  1  +3  +  6 +10  =  20  (4). 


^^'ob                              E 

LEMENTS    OF    ALGEBRA. 

[CHAP,  vni 

Now,  by  a  series  o 

f  subtractions  we  have 

Equation  (2)  — (1),  giv 

as   B  +  3C  +    7D  = 

3  .  .  .  (5), 

"          (3)-(2),      " 

5  +  5C+  19Z)  = 

6  .  .  .  (6), 

"          (4)-(3).       ' 

B  +  7C  +37D  = 

10  .  .  .  (7), 

"          (^^)-(5),       ' 

2C  +  12D  = 

3  .  .  .  (8), 

"         (7)-(6),      ' 

2C+  18D  = 

4   .   .  .  (9), 

"          (9)-(8),      ' 

6D  = 

1  ;  hence,  D  = -- ; 
o 

also, 

2C+  18D  = 

4,    gives      C  =  —  : 
>    to                       2  ' 

B  +  7C-\-37D  = 

10.      "        B  =  i-; 

A-\-B^  C  +  D  = 

1,        "         ^==0. 

Hence, 

1+3  +  6+10  + 

n(n+  1) 

}n  +  -^«2  +  -l«3 

= 

-^  (2  +  3«  +  n2) 

n(n +  !)(«  + 2) 

1.2.3" 

Let  us  suppose  that  we 
have  a  pile  of  balls  whose 
base  is  a  square,  two  sides 
of  which,  EF,  FH,  are  seen 
in  the  figure,  and  that  it 
terminated  by  a  single  ball 
at  G. 

Now,  the  number  of  balls 
in  the  upper  course  will  be 
expressed  by  P,  in  the  sec- 
ond course  by  2^,  in  the 
third  course  by  3^,  &c.     Hence,  the  series 

12  +  22  +  32  +  42  +  52  -r    .   .   .  n2, 
will    express  the  number  of  balls  in   a  square    pile,  of  Avhich  the 
lumber  of  courses,   and  consequently  the  number  of  balls   in  one 
of  the  lower  rows  is,  n. 

To  find  the  sum  of  this  series,  assume 

1  +  4  +  9  +  16  +    .    .   .   ifl  =  A  -\-  Bn  +  Cii^  -\-  Dn\ 


CHAP.   VIII.] 


SUMMATION    OF    SERIES. 


257 


from  which  we  find 

A+B+C-{-D  =  l  =    1  , 

A -{-2B  +    4C+    8D  —  1  +  4  =    5 

A  +  3B  +    9C  +  27D  =1  +  4  +  9  ^14 

A-\-  4B  +  IGC  +  64D  =  1+4  +  9+  16  =  30 

and  from  these  four  equations,  we  find,  by  continued  subtractions 


D  =  h     C  =  i,    B 


and    A  =  0  ;    hence, 


1+4  +  9+16  +  25 7*2  =  -J-«  +  J-?i2  +  inS 

=  ^{2n"  +  3n+  1) 

^n(n  +  \){2n  +  1) 
~  1  .     2      .         3 

Let  us  now^  suppose  that  we  have  a  rectangular  or  oblong  pile 
of  shot,  as  represented  in  the  figure  below. 


Suppose  we  take  off  from  the  oblong  pile  the  square  pile  EFD. 
We  then  see  that  the  oblong  pile  may  be  formed  by  adding  to 
the  square  pile  a  series  of  triangular  strata,  each  containing  as  many 
balls  as  are  contained  in  one  of  the  faces  of  the  square  pile  ;  and 
the  number  of  the  triangular  strata  -will  be  one  less  than  the  num- 
ber of  balls  in  the  top  row.  Therefore,  if  n  denote  the  number  of 
horizontal  courses,  the    number  of  balls    in    one    triangular    strata 

n(n  +  \) 

will  be  expressed  by ;    and  if  m+1   denotes  the  whole 

til 

number  of  balls    in    the    top  row,  the  number  of  triangular  strata 

will  be  denoted  by  m\  and  the  number  of  balls  in  all  these  strata 

bv 

X  m. 


17 


258  ELEMENTS    OF    ALGEBRA.  [CHAP.  VIII. 

But  since  the  number  of  balls  in  a  square  pile,  whose  side  con- 
tains n  balls  is 

n{n+  l)(2n+  1) 

1  .     2  ~         3      ' 
the  number  of  balls    in    an    oblong   pile,  whose   top  row  contains 
m  +  1  balls,  and  depth  n  balls,  will  be  expressed  by 

w(»  +  1)  (2ra  +  1)        n  {n  +  1) 
1.2.         3         +  2  ^  "^ 

_  n{n-\-\)        (1  +  271  +  3to) 

-         2  ^  ■  3  • 

If  we  denote  the  general  sum  by  S,  we  shall  have  the  follow 
ing  formulas  for  the  number  of  shot  in  each  pile. 

Triangular,    S  =  ~^^  \^—L-J.  =  — .     ^^    ^  („  +  i  +  i) 

o                  ^       n  (n  +  \)    (2n  +  1)        1      n(n+l)     ,      . 
Square,      S  =  -A_ L  .-^-g^— -'  =  -j .  -^g"—  •(«  +  «+  1). 

Rectangular, 

n(n+l)    (2n+l+3m)      1     n{n  +  l) 
*  — —- 2 • g- =  3  • — 2 -l{n+m)  +  {m  +  n)  +  {mi-\)]. 

Now,  since is  the  number  of  balls  in  the  triangular 

face  of  each  pile,  and  the  other  factor,  the  number  of  balls  in  the 
longest  line  of  the  base  plus  the  number  in  the  side  of  the  base 
opposite,  plus  the  parallel  top  row,  we  have  the  following 

RULE. 
Add  to  the  number  of  balls  in    the    longest    line  of  the  base,  the 
number  in  the  parallel  side  opposite,  and  also  the  number  iji  the  top 
parallel  row;  then  multiply  this  sum  by  one  third  the  number  in  tri- 
angular face. 

EXAMPLES. 

1.  How  many  balls  in  a  triangular  pile  of  15  courses? 

Ans.  G80 

2.  How  many  balls   in  a  square  pile  of   14   courses?    and  how 
many  will  remain   after   5   courses   are  removed  ? 

Ans.   1015  and  960. 


CHAP.   VIII.]  SUMMATION    OF    INFINITE    SERIES.  259 

3.  In  an  oblong  pile  the  length  and  breadth  at  bottom  are  respect- 
ively 60  and  30  :  how  many  balls  does  it  contain  ?       Ans.  23405. 

4.  In  an  incomplete  rectangular  pile,  the  length  and  breadth  at 
bottom  are  respectively  46  and  20,  and  the  length  and  breadth  at 
top  35  and  9:  how  many  balls  does  it  contain?         Ans.  7190 

Summation  of  hijinite  Series. 

247.  An  infinite  series  is  a  succession  of  terms  unlimited  in 
number,  and  derived  from  each  other  according  to  some  fixed  and 
known  law. 

The  summation  of  a  series  consists  in  finding  an  expression  of 
a  finite  value,  equivalent  to  the   sum  of  all  its  terras. 

Different  series  are  governed  by  different  laws,  and  the  methods 
of  finding  the  sum  of  the  terms  which  are  applicable  to  one  class, 
will  not  Apply  universally.  A  great  variety  of  useful  series  mav 
be  summed  by  the  following  formula  : 

Assume  — 7 —  =  — 7^^— — r : 

n        n  -\-  p       n  [n  -{-  p) 

then,  ^—-1.(1. ?_). 

n  [n  -\-  p)        p   \n        n  -\-  p/ 

If  now,  by  attributing   known  values  to  p  and   q,  and    different 

values  in  succession  to  n,  the  expression   — ; r    shall    repre- 

n{n-\-  p) 

sent  a  given  series  ;    then,  the    sum  of  this    series   will   bo   equal 

(0    —   multiplied  by  the   difference    between    the    two   new  series 
P 

of  which  —  and  — - —  are  the  general  terms.     Hence,  if  the  dif- 
n  n  -\-  p 

ference  of  the  sums  of  these  series  be  known,  and  the  value  of 

—  be  known,  we  can  find  the  value  of   the   series  — ; — ; — -,  by 
p  n{n+p) 

the  formula  *S  =  —  (s'  —  s")  even   if  we    do    not   know  the  value 
P 

of   he  now  series  —  and 


n  n  -\-  p 

EXAMPLES. 

1     Required  the  sum  of  the  series 

1 1 1 \-  &c.,  to  infinity 

1.2       1.2.3       1  .3.4^  1.4.5  ' 


260  ELEMENTS    OF    ALGEBRA.  [CHAP.   VIII 

We  see  that  if  we  make  q  =  I,  and  p  =  1,  and   n  =  1,  2,  3 
4,  &c.,  in  succession,  that  the  first  member  of  the  formula, 

9 

n  (tj  +  pY 

will,  in  succession,  represent  each  term  of  the  series ;  while  un 

(ler  the  same  supposition,  the    second  member  will  become,  for  w 

terms  of  the  series. 


111  1 

'+T  +  T  +  T---    T 

L         \2^34  n+l/_ 


'  71+1        n  +  1 


If  now,  we  suppose   n  =  oo,  the  value  of  the  sum  of  the  series 
will  become  equal  to  1. 

2.  Required  the   sum  of  n  terms   of  the   series 

To  adapt  the  formula  to   this    series,  we    make    q  =  I,  p  =:  2, 
and  n  =  1,  3,  5,  7,  &c.;   we  then  have,  for  the  sum  of  n  terms, 

111  1 

1  +-7r  +  -^  +  ^ 


3         5         7  2tt  —  1 

=  1 


-  (y  +  -, 


+  -=r  •      7: r  + 


1  _    1         •         1         i  2w  +  1 

7    '      2 
2n  ,      1 


5         7  2«  —  1       2/1  +  1 


-,    and    —    of  this  sum    = 


2ra  +  r  p  211+1 

If  now,  we    suppose  n  =  oo,  the  value  of  the    series   becomes 
equal  to  one  half. 

3.  Required  the  sum  of  n  terms  of  the  series 

-\ 1 1 1-  &c.,  to  infinity. 


1.4    '     2.5    '     3.6   '     4.7 
Here  p=  3,  q  =  I,  n  =  1,  2,  3,  4,  &c. :    hence, 


1 

—  < 
3 


111  1 

2  3         4  n 


1 

+  1    '    n  +  2    '    n  +  3  . 


/I  1  1  1.1.1. 

\4         5  6  71         n  -\-  I        n  -i 

=  1  rii  -  r-i-  +  -—  +  -^)  1  =  - 

3  1-6         \n  +  1  ^n  +  2  ^«  + 3/J       18' 
when   n  =  (30. 


CHAP.  VIII  ]  SUMMATION    OF    INFINITE    SERIES.  2fil 

4.  Find  the  sum  of  n  terms  of  the  series 

2  3  4  5  6  ,  . 

3-.T  -  577  +  779  "  9^1  +  n:i3  +  ^'^•'  '"^  '"^"^'>' 

r    2  3  4_       _5_  _    n+l 

I   "3         b        'Y  ~~^  '^   '  '  '  '    "^  2ra+  1 


[  -( 


2_        34_  ^        n        _„-|-l 

5"  "  y       "9"  "    ■   ■   '   ■    ~  2n  4-  1  ^  2;i  +  3 


which  becomes, 

2  n+  1 

T  ~  2«  +  3 


(1  _1  4-  1  _ ±  1). 


If  the  number  of  terms  used  is  even,  the  upper  siwn  will  apply, 
the  quantity  within  the  parenthesis  will  become  +  1,  and  the  sum 
of  the  n  terms  before  dividing  by  p,  is 

1  n  -I-  1         1 

1 =  — -,    wtien    1  =  CE. 

3        2n  +  3        6 

J f  71  is  odd,  the  lower  sign  is  used,  and  the  quantity  within  the 
parenthesis  reduces  to  zero,  and  we  have 

2  n+\         1 
=  — -,    when    n  =  oo. 

3  2«  +  3        6 

Then,  since  p  =2,  the  sum  of  the  series  when  n  =:  oo,  is  ---. 

5.  Required  the  sum  of  the  series 

4  4  4  4  4 

1 1 1 1 h  &c.,  to  infinity. 

1    5  ^  5.9  ^  9.13  ^  13.  17   ^   17  .21  ^  ^ 

Ans    ] 


262  ELEMENTS    OF    ALOEBRA.  [CHAP.  IX. 


CHAPTER  IX. 

CONTINUED    FRACTIONS,    EXPONENTIAL    (JUANTITIES,    LOGARITHMS, 
AND    FORMULAS    FOR    INTEREST. 

248.  Every  expression  of  the  form 

1  1  1 


1  .1  .         1 

a  +  -r,         a  -\ r  '^  + 


in  which  a,  b,  c,  d,  &c.,  are  positive   whole  numbers,  is  called  a 
continued  fraction. 

Hence,  a  continued  fraction  has  for  its  numerator  the  unit  1,  and 
for  its  denominator  a  whole  number,  plus  a  fraction  which  has  1  for 
its  numerator  and  for  its  denominator  a  whole  number  plus  a  frac- 
tion, and  so  on. 

249.  The  resolution  of  an  equation  of  the  form 

gives   rise   to   continued   fractions.     Suppose  for  example,   a  =  8, 
6  =  32.     We  then  have 

8^  =  32, 
in  which   a;  >  1,    and   x  <^2.     Make 

a;  =  1  +  — , 

y 

in  which   y  >  1,  and  the  proposed  equation   becomes,  after  chan- 
ging the  members, 

32  =  8*    y  =  8  X  8y,     whence, 

\_ 
82'  =  4     and  consequently,    8  =  4". 


CHAP.   IX.]  CO.NTINUED    FRACTIONS.  263 

It  is  plain,  that  the  value  of  y  lies  between  1   and  2.     Suppose 

1 

i+i  - 

and  we  have,  8=4     ^=4x4^; 

\_ 

hence,  4^=2,    and    4  =  2^    or    2  =  2 

113 

But,  y=l+_^14-_=:_; 

1  1  2         5 

and  a:=:l+  —  z=l+  —  =1  +  —  =  —  ; 

y  2  3         3 

and  this  value  will  satisfy  the  proposed  equation.     For, 

8^  =  83"  =  V"P  =  V(2^  =  yW?  =  2^  =  32. 
250.  If  we  apply  a  similar  process  to  the  equation 
10^  =  200, 
we  shall  find 

1  1  1 

a=2H ;     y  =  3H ;     s  =  3-| . 

y  z  u 

Since  200  is  not  an  exact  power,  a;  cannot  be  expressed  either 
by  a  whole  number  or  a  fraction  :  hence,  the  value  of  x  will  be 
incommensurable,  and  the  continued  fraction  will  not  terminate,  but 
will  be  of  the  form 

1  I  1 

x  =  2H ==2H =  2  + 


"  3  +  i  3+        ' 


^  o  1 

u  +  <vc. 
251.  Common  fractions   may  also  be  placed  under  the  form  of 
continued  fractions. 

65 
Let  us  take,  for  example,  the  fraction    — — ,    and  divide  both  its 

terras  by  the  numerator  65,  the  value  of  the  fraction   will  not   be 
changed,  and  we  shall  have  \ 

65   _T49 
149  ~"     65' 

65  1 


or  effecting  the  division, 


149                19 
2  H . 


264  ELEMENTS    OF    ALGEBRA..  [CHAP.   IX. 

19 

Now,  if  we  neglect  the  fractional  part    —    of  the   denominator, 

we  shall  obtain    -^   for  the  approximate  value  of  the    given    frac- 

tion.     But  this  value  would   be   too   large,  since    the    denominator 

used  was  too  small. 

19 
If,  on  the  contrary,  instead  of  neglecting  the  part   -— ,  we  were 

to   replace   it   by  1,    the   approximate   value  would   be    -^,    which 

would  be  too  small,  since  the  denominator  3  is  too  large.     Hence. 

65         1  ,      65         1 

- —  <"  —     and     >  — , 

149        2  149        3 

1  •    1 

therefore  the  value  oi  the  fraction  is  comprised  between  -—  and  — . 

If  we  wish  a  nearer  approximation,  it  is  only  necessary  to  op- 

,         n  ■  19  11        '        1  •  /.  •  65 

erate  on  the  fraction    —    as  we  did   on    the    given  traction 


65  °  149 

and  we  obtain 


19 

1 

65 

-S' 

65 

1 

149 

^+19 

hence, 


If  now,  we  neglect  the  part   — ,  the  denominator  3  will  ht-  letss 

than  the  true  denominator,  and   -7-    will  be   larger  than    the  nuin- 

1 

ber  which  ought  to  be    added  to  2  ;    hence,  1   divided  by   2  H 

will  be  less  than  the  true  value  of  the  fraction  ;  that  is,  if  we  stop 
at  the  first  reduction  and  omit  the  fractional  numbers,  the  result 
will  be  too  great ;  if  at  the  second,  it  will  be  too  small,  &c.  Hence, 
generally,  if  we  stop  at  an  odd  reduction,  and  neglect  the  fractional 
part,  the  result  will  be  too  great ;  but  if  we  stop  at  an  even  reduc- 
tion, and  neglect  the  fractional  part,  the  result  will  be   too   small 


CHAP.  IX.]  CONTINUED  FRACTIONS  265 

Making  two  more  reductions  iu  the  last  example,  we  have 
65  1 


149  1 

2H ....     1st  reduction,  too  great;' 

3H -     2d  "  too  small; 

2  H 3d  "  too  great ; 

2  H 4th  "  too  small ; 

1  4-  —  5th  "  too  great. 

252.  The  separate  fractions 

1  1  1 


a  \  1 

«  4-  -r.         a  + 


6'  '1 

c 
are  called  approximating  fractions,  because    each    affords,  in    suc- 
cession,  a  nearer  value  of  the  given  expression. 

The   fractions    — ,    — -,    — ,  &c.,  are  called  integral  fractions, 
a        0       c  o       J 

When  the  expression  can  be  exactly  expressed  by  a  vulgar  frac- 
tion,   as    in   the    numerical    examples    already   given,    the    integral. 

fractions    — ,    -— ,    — ,  &c.,  will  terminate,  and  we  shall  obtain  an 
a        0       c 

expression    for  the    exact    value    of  the    given    fraction   by   taking 

them  all. 

We  will  now  explain  the  manner  in  which  any  approximating 
fraction  may  be  found  from  those  which  precede  it. 

1.  —     ----=  —  1st  app.  fraction. 

2.     —  -     -      -      -       =       — — ; — -         2d   app.  fraction 

1  flO  +  1 

'^  +  T 

3.     -      -       =  7—. 'r-^ 3d    app.  fraction 

1  (ab  +  1  Ic  +a 


a 

b 

ab  +  1 

bc-\-  1 

a  + 


b  + 


{ab  +  l)c  +a 

T 

c 

By  examining  the  third  approximating  fraction,  we  see,  that  its 
numerator  is  formed  by  multiplying  the  numerator  of  the  prece- 
ding fraction  by  the  denominator  of  the  third  integral  fraction,  and 


266  ELEMENTS    OF    ALGEBRA.  [CHAP.   IX 

adding  to  the  product  the  numerator  of  the  first  approximating 
fraction  :  and  that  the  denominator  is  formed  by  muhiplying  the 
denominator  of  the  last  fraction  by  the  denominator  of  the  third 
integral  fraction,  and  adding  to  the  product  the  denominator  of 
the   first  approximating  fraction. 

We  should  infer,  from   analogy,  that   this    law   of  formation   is 

general.     But  to  prove  it  rigorously,  let   — ,  — ,  — ,  be  any  three 

approximating  fractions  for  which  the  law  is  already  established. 
Since  c  is  the  denominator  of  the  last  integral  fraction,  we  have 
from  what  has   already  been  proved, 

Let   us   now  add  a  new   integral  fraction   —    to    those    already 

deduced,  and  suppose    -^   to  express  the  next  approximating  frac- 

P  S 

tion.     It  is  plain  that   -=r    ^^^^  become    —   by  simply  substituting 

1       , 
tor  c,    c  -f — -  :    hence, 
a 

S  _^V  '^t)"^^  _  {Qc-\-  P)d  +  Q   _  Rd-\-  Q 

Hence,  we  see  that  the  fourth  approximating  fraction  is  deduced 
from  the  two  immediately  preceding  it,  in  the  same  way  that  the 
third  was  deduced  from  the  second  and  first ;  and  as  any  fraction 
may  be  deduced  from  the  two  immediately  preceding  in  a  similar 
manner,  we  conclude  that,  the  numerator  of  the  n^*^  approximating 
fraction  is  formed  hy  multiplying  the  numerator  of  the  preceding 
fraction  by  the  denominator  of  the  n^**  integral  fraction,  and  adding 
to  the  product  the  numerator  of  the  n  —  2  fraction ;  and  the  denom- 
inator is  formed  according  to  the  same  law,  from  the  two  preceding 
denominators. 

253.  If  we  take  the  difference  between  any  two  of  the  con- 
secutive approximating  fractions,  we  shall  find,  after  reducing  them 
to  a  common  denominator,  that  the  difference  of  their  numerators? 


CHAP.  IX.]                               CONTINUED    FRACTIONS.  267 

will  be  equal  to  ±  1 ;  and  the  denominator  of  this  difference  will 
be  the  product  of  the  denominators  of  the  fractions. 

Taking,  for  example,  the  consecutive  fractions  — ,  and  — r -, 

a  ao  -\-  I 

we  have 

1  b  ab+l  —ab  +1 


and 


a        ab  +  1  a(ab-{-l)         a{ab  +  1)' 

b  bc+  }  —  1 


ab  +  1        (ab  +  1)  c  -\-  a  ~  {ab  -i-  1)  [{ab  +  1  )  c  +  aj 
To  prove  this  property  in  a  general  manner,  let 

P_    Q_   R^ 

R'    Q"   R'' 

be  three  consecutive  approximating  fractions.     Then 
P       Q       PQ'-P'Q 


and 


P'      Q'  P'Q' 

Q       R       R'Q  -  R( 


Q'      R'  Q'R' 

But  Rz=iQc-\-  P   and   R'  z=z  Q'c  +  P'  (Art.  252). 

Substituting  these  values  in  the  last  equation,  we  have 
Q        R        {Q'c  +  P')Q-{Qc-\-P)i:r 


or  reducing 


Q' 

R'~ 

R'Q' 

Q 

R 

P'Q~  PQ' 

Q' 

R'  ~ 

R'Q' 

From  which  we  see,  that  the  numerator  of  the  difference  -p^  —  ^ 


Q 

Q 

Q      R 


is  equal,  with  a  contrary  sign,  to  that  of  the  difference  . 

Q'  R' 
That  is,  thf  difference  between  the  numerators  of  any  two  consecu- 
tive approocima.'ing  fractions,  when  reduced  to  a  common  denominator, 
is  the  same  with  a  contrary  sign,  as  that  yahich  exists  between  the 
last  numerator  and  the  numerator  of  the  fraction  immediately  fol- 
lowing. 

But  we  have  already  seen  that  the  difference  of  the  numerators 
of  the  1st  and  2d  fractions  is  equal  to  +  1  ;  also  that  the  differ- 
ence between  the  numerators  of  the  2d  and  3d  fractions  is  equal 
to  —  1  ;  hence,  the  difference  between  the  numerators  of  the  3d 
and  4th  is  equal  to   +  1  ;    and  so  on  for  the  following  fractions 


268  ELEMENTS  OF  ALGEBRA  [CHAP.  IX. 

Since  the  odd  approximating  fractions  are  all  greater  than  the 
true  value  of  the  continued  fraction,  and  the  even  ones  all  less 
(Art.  251),  it  follows,  that  when  a  fraction  of  an  even  order  is 
subtracted  from  one  of  an  odd  order,  the  difference  should  have 
a  plus  sign ;  and  on  the  contrary,  it  ought  to  have  a  minus  sign, 
when  one   of  an  odd  order  is  subtracted  from  one  of  an  even. 

254.  It  has  already  been  shown  (Art.  251),  that  each  of  the 
approximating  fractions  corresponding  to  the  odd  numbers,  exceeds 
the  true  value  of  the  continued  fraction  ;  while  each  of  those  cor- 
responding to  the  even  numbers,  is  less  than  it.  Hence,  the  dif- 
ference between  any  two  consecutive  fractions  is  greater  than  the 
difference  between  either  of  them  and  the  true  value  of  the  con- 
tinued fraction.  Therefore,  stopping  at  the  n'^  fraction,  tht;  result 
will  be  true  to  within  1  divided  by  the  denominator  of  the  7t"' 
fraction,  multiplied  by  the  denominator  of  the  fraction  which  fol- 
lows. Thus,  if  Q'  and  R^  are  the  denominators  of  consecutive 
fractions,  and   we   stop  at  the   fraction   whose   denominator   is    Q', 

the  result  will  be   true   to  within    ■z^;-^}-     But  since  a,  b,  c,  d,  &c., 

are  entire  numbers,  the  denominator  R^  will   be  greater  than    Q' 

and  we  shall  have 

1  1 

nence,  if  the  result  be    true  to  within  ,    it  will  certainly  be 

true  to  within  less  than   the  larger  quantity 

1 


Q'2  ' 

•hat  is,  the  approximate  result  which  is  obtained,  is  true  to  within 
unity  divided  by  the  square  of  the  denominator  of  the  last  approxi- 
mating fraction  that  is  employed. 

If  we  take  the  fraction ,    we  have 

347' 

829  1 

=  2  + , 


347 

3  + 


CHAP     IX.]  EXPONENTIAL    QUANTITIES.  269 

Here,  we  have  in  the  quotient  the  whole  number  2,  which 
may  either  be  set  aside,  and  added  to  the  fractional  part  after  its 
value  shall  have  been  found,  or  we  may  place  1  under  it  for  a 
denominator,  and  treat  it  as   an  approximating  fraction. 

Resolution  of  the  Equation  a'  =  h. 

255.  An  equation   of  the  form 

0^=  h, 
is    called   an    exponential  equation.     The    object   in    resolving   tliis 
equation  is,  to  find  the   exponent  of  the  power  to  which  it  is  ne- 
cessary to  raise  a  given   number   a,  in  order  to   produce    another 
given  number  b. 

Suppose  it  were  required  to  resolve   the  equation 

2^  =  64. 
By  raising   2    to    its    different   powers,    we    find  that    2^  =  64 ; 
hence,    a;  =  6    will  satisfy  the   conditions   of  the   equation. 
Again,  let  there  be   the   equation 

3*  =  243,     in  which     x  —  b. 
In  fact,   so  long    as   the   second  member  Z»  is  a  perfect  power  of 
the   given  number  a,  it  may  be   obtained  by  raising   a  to  its  sue 
cessive  powers,  commencing  at  the  first. 

Suppose  it  were  required  to  resolve  the  equation 

2^=  6. 
By  making   x  =  2,   and   x  =  3,   we  find 
22  =  4    and   2^  =  8  ; 
from  which  we  perceive  that  the  value  of  x  is  comprised  between 
2  and  3. 

Make  then,  x  =  2  -\ -,    in  which    a/  >  1 . 

ar 

Substituting  this  value  in  the  given  equation,  it  becomes, 
2     '='  =  6,     or     22x2^  =  6;     hence 


—        6         3 

ox/  _  _^ _ . 

^      -   4   -  2' 

and  by  changing    the 

terms     and   raising   both  members    to  the 

a/  power, 

\2/ 

270  ELEMENTS    OF    ALGEBRA.  [CHAP.  IX. 

To  determine  x\  make  successively  3/  =.\  and  2  ;    we  find 

therefore,  a^  is  comprised  between  1  and  2. 

Make,  of  =  \  -{ ,    in  which    a;''  >  1. 

x' 

(  3  \^' 
By  substituting  this  value  in  the  equation     (— j    =  2, 

{\Y^=^:     hence,     |  x  (|)^^  =  2, 

and  consequently,  (  — )     =  — . 

4         3 

The  hypothesis        x"=.\,     gives     — < — ; 

and  of  x"  =  2,     gives     -r-  >  — ; 

.     b  9^2 

therefore,  x'^  is  comprised  between  1  and  2. 

Let  a/'=  I  -\ —  ;    then, 

4\i+-^       3        ,  4         /4\-^,       3 


(-)    .'"=-;     hence,    -  x  (-)""=  ^ 

.QvV//  4 

Whence,  [-)      = -. 


9  \^'^'       4 

If  we  make  x"^  —  2,  we  have 

9\2       81         4 


Vs)  ~  "eT  ^  "3"' 

and  if  we  make  x^'''  =  3,  we  have 

/9\3_  729        4  _ 

\T/  ~  5T2     T ' 

therefore,  a/^^  is  comprised  between  2  and  3, 

Make  a^^'  =  2  -| ■,  and  we  have 

/9\2+-!-       4        -  81/9\-i-       4 

{_)     siy= — ;     hence,    —  (  — )x>v=: — . 
\8/  3  '64\8/  3 

/256\it'v       9 

and  consequently,  1'243"/      ~  "s"" 


CHAP.   IX.]  EXPONENTIAL    QUANTITIES.  271 

Operating  upon  this  exponential  equation  in  the  same  manner 
as  upon  the  preceding  equations,  we  shall  find  two  entire  num- 
bers, k  and  h-\-  \,  between  which  x^^  will  be  comprised. 

Making 

1 

and  x^  can  be  determined  in  the  same  manner  as  a;'^,  and  so  oi;. 
Making  the  necessary  substitutions  in  the  equations 

X  xf  X  ■  a;'^ 

we  obtain  the  value  of  x  under  the  form  of  a  continued  fraction 

x=l1-\ 

1 

^+ 1 

1  +  - 


1 

2H 


Hence,  we  find  the  first  three  approximating  fractions  to  be 
1       _!_       3 

T'  ¥'  y 

and  the  fourth  is  equal  to 

l2il±i=l(A„.2K). 
5x2  +  2       12  '^  '' 

which  is  the  true  value  of  the  fractional  part  of  x  to  within 
^'^    -T7T  {^^t-  254). 


(12)2  144 

Therefore, 

7        31  1 

x  =  2  +  —  —  —  =:  2.58333  +     to  within 


12       12         ■  144  ' 
and  if  a  greater  degree  of  exactness  is  required,  we  must  take  a 
greater  number  of  integral  fractions. 

EXAMPLES. 

X  =         2.46    to  within  0.01. 

0;=          0.477  "          0.001. 

X-    ~   0.25  "          0.01 


3* 

= 

15 

10* 

z=. 

3 

5* 

= 

2 
3 

272  ELEMENTS    OF    ALGEBRA.  [CHAP.   IX 

Theory  of  Logarithms. 

256.  If  we  suppose  a  to  preserve  the  same  value  in  the  equation 

a^  —  N, 
and  N  to  become,  in  succession,  every  possible  positive  number,  il 
is  plain  that  x  will  undergo  changes  corresponding  to  those  made 
in  N.  By  the  method  explained  in  the  last  Article,  we  can  de- 
termine, for  each  value  of  N,  the  corresponding  value  of  x,  either 
exactly  or  approximatively. 

Any  number,  except  1,  may  be  taken  for  the  invariable  num- 
ber a ;  but  when  once  chosen,  it  is  supposed  to  remain  the  same 
for  the  formation  of  one  entire  series  of  numbers. 

The  exponent  x  of  a,  corresponding  to  any  value  of  N,  is  called 
the  logarithm  of  that  number ;  and  the  invariable  number  a  is  called 
the  base  of  that  system  of  logarithms.     Hence, 

The  logarithm  of  a  number,  is  the  exponent  of  the  power  to  which 
It  is  necessary  to  raise  an  invariable  number,  called  the  base  of  the 
system,  in  order  to  produce  the  number. 

The  general  properties  of  logarithms  are  independent  of  any 
particular  base.  The  use  that  may  be  made  of  them  in  nu- 
merical calculations,  supposes  the  construction  of  a  table,  con- 
taining all  the  numbers  in  one  column,  and  the  logarithms  of 
these  numbers  in  another,  calculated  from  a  given  base.  Now,  in 
calculating  this  table,  it  is  necessary,  in  considering  the  equation 

a^  =  N, 
to  make  N  pass   through  all  possible  states  of  value,  and  to   de 
termine  the  value  of  x  corresponding  to   each  of  the  values  of  N, 
which  may  be  done  by  the  method  of  Art.  255. 

257.  The  base  of  the  common  system  of  logarithms,  or  as  they 
are  sometimes  called,  Briggs'  logarithms,  from  their  inventor,  is 
uhe  number  10.  If  we  designate  the  logarithm  of  any  number 
by  log.  or  I,  we  shall  have 

(10)0=  1-  hence, 

(10)1=         10;  hence, 

(10)2=       100;  hence, 

(10)3=     1000;  hence, 

(10)*=  10000;  hence, 
&c., 


log. 

1 

=  0; 

log. 

10 

=  1; 

log. 

100 

=  2, 

log. 

1000 

=  3 

log. 

10000 
&c. 

=  4. 

CHAP.  IX.] 


THEORY  OF  LOGARITHMS. 


273 


Hence,  in  the  common  system,  the  logaruhm  of  any  numt/cr 
between  1  and  10,  is  >  0  and  <  1.  The  logarithm  of  any  nu!;;- 
ber  between  10  and  100,  is  >  1  and  <  2;  the  logarithm  of  any 
nmtiber  between    100   and   1000,   is    >  2   and   <  3  ;    and  so  on. 

Hence,  the  logarithm  of  any  number  expressed  by  two  figures. 
and  which  is  not  a  perfect  power  of  the  base  of  the  sysleni. 
will  be  equal  to  a  whole  number  plus  an  approximating  fraction, 
the  approximate  value  of  which  fraction  is  generally  expressed 
decimally. 

The  integral  part  of  a  logarithm,  is  called  the  index  or  char- 
acteristic of  the  logarithm. 

By  examining  the  several  powers  of  10,  we  see,  that  if  a  num- 
ber is  expressed  by  a  single  figure,  the  characteristic  of  its  logarithm 
will  be  0 ;  if  it  is  expressed  by  two  figures,  the  characteristic  of 
its  logarithm  will  be  1  ;  if  it  is  expressed  by  three  figures,  the 
characteristic  will  be  2  ;  and  if  it  is  expressed  by  n  places  of 
figures,  the  characteristic  will  be  n  —  ]    units. 

The  following  table  shows  the  logarithms  of  the  numbers,  from 
1   to  100. 


N. 
1 

Log. 
0.000000 

N. 
26 

Log.    ] 
1.414973  1 

N. 
51 

hog. 

N. 
76 

Log. 

1.707570 

1.880814 

2 

0.301030 

27 

1.431364 

52 

1.716003 

77 

1.886491 

3 

0.477121 

28 

1.447158  i 

53 

1.724276 

78 

1.892095 

4 

0.602060 

29 

1.46-2398 

54 

1.732394 

79 

1.897627 

5 
"6 

0.698970 
0.778151 

30 
3l 

1.477121  1 
1.491362  1 

55 
56 

1.740363 

80 
87 

1.903090 

1.748188 

1.908485 

7 

0.845098 

32 

1.505150  1 

57 

1.755875 

82 

1.913814 

8 

0.903090 

33 

1.518514  i 

58 

1.763428 

83 

1.919078 

9 

0.954243 

34 

1.531479  j 

59 

1.770852 

84 

1.924279 

'  10 

1.000000 

35 

1.544068 

60 

1.778151 

85 

1.929419 

IT 

1.041393 

36 

1.556303 

6] 

1.785330 

86 

1.934498 

12 

1.079181 

37 

1.568202 

62 

1.792392 

87 

1.939519 

13 

1.113943 

38 

1.579784 

63 

1.799341 

88 

1.944483 

14 

1.146128 

39 

1.591065 

64 

1.806180 

89 

1.949390 

15 

1.176091  j 

40 

1.602060  I 

65 

1.812913 

90 

1.954243 

16 

1.204120 

41 

1.612784  i 

66 

1.819544 

91 

1.959041 

!  17 

1.230449 

42 

1.623249 

67 

1 .826075 

92 

1.963788 

18 

1.255273 

43 

1.633468 

68 

1.832509 

93 

1.968483 

19 

1.278754 

44 

1.64345:^ 

69 

1.838849 

94 

1.973128 

20 
2T 

1.301030 
1.322219 

45 
46 

1.653213 

70 
7l 

1  845098 
1.8.512.58  i 

95 
96 

1.977724 

1.662758 

1.982271 

22 

1.342423  1 

47 

1.672098  ■ 

72 

1.857333 

97 

1.986772 

23 

1.361728 

48 

1.68I24I 

73 

1.863323 

98 

1.991226 

I  24 

1.380211 

49 

1.690196 

74 

1.869232 

99 

1.995635 

i  25 

1.397940  ! 

50 

1.698970  1 

75 

1.875061 

100 

2.000000 

274  ELEMENTS    OF    ALGEBRA,  [CHAP.  IX. 

The  chavacterislic  being  always  one  less  than  the  number  of 
places  of  figures  in  the  number,  is  not  written  down  in  the  table 
of  logarithms  for  numbers  which  exceed  100.  Thus,  in  search- 
ing for  the  logarithm  of  2970,  we  should  find  in  tho  table  oppcsite 
2970,  the  decimal  part  .472756.  But  since  the  number  is  ex- 
pressed by  four  figures,  the  characteristic  of  the  logarithm  is  3. 
Hence,  log.  2970  =  3.472756, 

and  by  the   definition  of  a  logarithm,  the   equation 
a^  =  N,     gives 

103.472756    _  2970. 

Multiplication  and  Division  by  Logarithms. 

258.  Let  a  be  the  base  of  a  system  of  logarithms,  and  sup- 
pose the  table  to  be  calculated.  Let  it  be  required  to  multiply 
together  a  series  of  numbers  by  means  of  their  logarithms.  De- 
note the  numbers  by  N,  N',  N" ,  N"\  &c.,  and  their  correspond- 
ing logarithms  by  .r,  x' ,  x" ,  x"\  &c.  Then,  by  definition  (Art. 
256),  we  have 

aF  —  N,      a"'  -  N\     a^"  =  A^'',     a^'"  =  N"'  .   .   .   &c. 
Multiplying    these  equations  together,  member  by  member,   and 
applying  the  rule  for  the  exponents,  we  have 

a^+x'+x^'+x^'^  .   .   .   =  N  X  N'  X  N"  X  W"      .   . 

But  since   a  is  the   base  of  the   system,  we  have 

x^  x'  ^x"  -^  x'"  .  .  .  =  log.  [N,  W,  N'\  N'"  ....); 

that  is,  the  sum  of  the  logarithms  of  any  number  of  factors,  is  equal 
to  the  logarithm  of  tlic  -product  of  those  factors. 

259.  Suppose  it  were  required  to  divide  one  number  by  another 
Let  N  and  iV^  denote  the  numbers,  and  x  and  x'  their  logarithms 
We  have  the  equations 

ax  —  N     and     ax'  =  N'  ; 

qX  jV" 

hence,  by  division  =  a^~x'  =:  -—-  ; 

/  N  \ 
or  X  -  y  =  log.  N-  log.  N'  =  log.  ^— j, 

that  is,  the  difference  between  the  logarithm  of  the  dividend  and  the 
logarithm   of  the  divisor,  is  equal  to   the  logarithm  of  the  quotient 


CHAP.  IX.]  THEORY    OF    LOGARITHMS.  275 

Consequences  of  these  Properties. 

A  multiplication  can  be  performed  by  taking  the  logarithms  of 
the  two  factors  from  the  tables,  and  adding  them  together  ;  this 
will  give  the  logarithm  of  the  product.  Then  finding  this  new 
logarithm  in  the  tables,  and  taking  the  number  which  corresponds 
to  it,  we  shall  obtain  the  required  product.  Therefore,  by  a  sim- 
ple addition,  we  jind  the  product  arising  from  a  mullipUcation. 

In  like  manner,  when  one  number  is  to  be  divided  by  another, 
subtract  the  logarithm  of  the  divisor  from  that  of  the  dividend, 
then  find  the  number  corresponding  to  this  difference ;  this  will 
be  the  required  quotient.  Therefore,  hy  a  simple  subtraction,  we 
obtain  the  quotient  arising  from  a  division. 

Formation  of  Powers  and  Extraction  of  Roots. 

260.  Let  it  be  required  to  raise  a  number  N  to  any  power  de- 

m 
noted  by    — .     If   a   denotes    the    base  of  the    system,  and  x  the 

n  '' 

logarithm  of  iV,  we  shall  have 

a^  =  N,     or     N  =z  a^; 

tn 
whence,  by  raising  both  members  to  the  power   — , 

n 

m  m 

TIT * 

iV"  =  a"    . 
Therefore,  log.  I  iV  "  )  =  —  .  a;  =  —  .  log.  N. 

If  we  make   ra  =  1  ;  there  will  result, 

m  .  log.  N  =  log.  iV"» ; 
an  equation  which  is  susceptible  of  the  following  enunciation  : 

If  the  logarithm  of  any  number  be  multiplied  by  the  exponent  of 
the  power  to  which  the  number  is  to  be  raised,  the  product  will  be 
equal  to  the   logarithm   of  that  power. 

261.  Suppose,  in  the  first  equation,  m  =  I  ;    there  will  result. 

1  — 

—  log.  N  =  log.  iV"  =  log.  V^;    that  is, 
n 

77(6  logarithm  of  any  root  of  a  number  is  obtained,  by  divt  ling 
(he  logarithm   of  the  number  by  the  index  of  the  root. 


276  ELEMENTS    OF    ALGEBRA.  [CHAP.  IX.. 

Consequences. 

To  form  any  power  of  a  number,  take  the  logarithm  of  this 
number  from  the  tables,  multiply  it  by  the  exponent  of  the  power ; 
then  the  number  corresponding  to  this  product  will  be  the  required 
power. 

In  like  manner,  to  extract  the  root  of  a  number,  divide  the  log- 
arithm of  the  proposed  number  by  the  index  of  the  root ;  then 
the  number  corresponding  to  the  quotient  will  be  the  required 
root.  Therefore,  by  a  simple  multiplication,  loe  can  raise  a  qua7i- 
tity  to  a  power,  and  extract  its  root  hy  a  simple  division. 

262.  If  we  make  the  different  exponents  of  10  negative,  the 
powers  corresponding  thereto  will  be  decimal  fractions.     Thus, 

(10)-'=     —     1=0.1;  hence,      log.0.1         =-1; 

(10)-2=:    -—    =0.01;         hence,      log.  0.01      =—2; 

Cl0)-3  =  -J—  =  0.001  ;       hence,      log.  0.001     =-3; 
>.     )  1000  »  '         o 

CIO)-*  = =  0.0001  ;     hence,      log.  0.0001  =  —  4. 

^     ^~        10000  ^ 

&c.,  &c.,  &c. 

The  logarithm  of  any  fraction  between  oi;e  and  one  tenth, 
as  four  tenths,  for  example,  may  be  expressed  thus, 

^''^'  (lo)  ^  ^''°'  (lo  ^  '^  )  ^  ^*'^"  To  '^  ^"^'"^  =  -  i  +  log.  4. 
For  the  fractions  between  one  hundredth  and  one  tenth,  as  six 
hundredths,  for  example,  we  have 

For  the  fractions  between  one  thousandth  and  one  hundredth, 
as  eight  thousandths,   for   example,  we  have 

log.  (-^)  =  log.  f-i—  X  8  )  =  —  3  +  log.  8. 
*=   VlOOO/  "   \1000  /  ° 

Now,  instead  of  performing  the  subtractions  indicated  above,  we 
unite  the  decimal  part  of  the  logarithm  to  the  negative  charac- 
teristic.    Thus, 

log.  0.4      =  —  1  +  log.  4  =:  —  1.G02060  ; 

log.  0.06    = —2  +  log.6  =  —2.778151  ; 

log.  0.008  =  —  3  +  log.  8  =  —  3.903090. 


CHAP.   IX.] 


THEORY    OF    LOGARITHMS. 


277 


Adopting  this  method  of  writing  the  logarithms,  we  see  that  the 
logarithm  of  a  decimal  fraction  may  be  found  from  the  tables,  by 
uniting  to  the  logarithm  of  its  numerator,  regarded  as  a  whole  num- 
ber, a  negative  characteristic  greater  by  unity  than  the  number  of 
ciphers  between  the  decimal  point  and  the  frst  signifcant  fgurc. 

To  demonstrate  this  in  a  general  manner,  let  a  denote  the  nii- 
[nerator  of  a  decimal  fraction,  and  b  its  denominator.  From  the 
nature  of  decimals,  we  shall  have 

b  =2  (lO)"*, 

in  which  m  will  denote  the  number  of  ciphers  in  the  denomina- 
tor.    Hence 

log.  — -  =  log.  \- — — )  zzz  log.  a  —  m  log.  10  r=  log.  a  —  m. 

*      6  *=     \(10)'"/  o  o  o 

Or,  in  other  words,  the  logarithm  of  a  whole  number  will  be- 
come the  logarithm  of  a  corresponding  decimal,  by  adding  to  it  a 
negative  characteristic  containing  as  many  units  as  there  are  ciphers 
in  the  denominator  of  the  decimal  fraction. 

Hence,  the  table  of  logarithms  whose  base  is  10,  will  give  the 
logarithms  of  all   decimals,   as  well   as   of  the   integral  numbers. 


GENERAL    EXAMPLES. 

1 .  What  is  the  square  of  7  ? 

Log.7 

Exponent  of  the  power 
Number  corresponding,  49 

2.  What  is  the  6th  power  of  2  ? 

W-2 

Exponent  of  the  power  - 
Number  corresponding,  64 

3.  What  is  the  cube   root  of  64  ? 

Log.  64  .... 

Then,    -         -         -         .         - 
Number  corresponding,  4 

4.  What  is  the  4th  root  of  81? 

5.  What  is  the  5th  root  of  32  ? 


=  0.845098 

z= 2_ 

17690196. 


0.301030 
6 

1.806180. 


=  1.806180 

3)  1.806180 

0.602060. 


Ans.  3. 
Ans.  2. 


278  ELEMENTS    OF    ALGEBRA.  [CHAP.  IX 

6.  Log.  (a  .  i  .  c  .  J  .  . .  .)  =  log.  a  -f  log.  b  +  log.  c  .  .  ,  . 

7.  Log.  f— - — )  =  log. a  + log.Z»4-log. c— log.(f— log. e. 

8.  Log.  (a'"  .b^.cP.  .  .  .)  =  m  log.  a  -\-  n  log.  b  -\-  p  log.  c  +  .  .  .  . 

9.  Log.  (a2_  a;2)  :=  log.  (a  +  a;)  +  log.  (a  —  x). 

10.  Log.  y(a2  _  a;2)  =  J  log.  (a  +  a;)  +  1  log.  (a  —  x). 

11.  Log.  (a^  X  V^)  =  3|log.  a. 

263.  Let  us  resume  the  general  equation 

and  suppose  a  to  be  the  base  of  a  system  of  logarithms.     Then, 
1st,  we  have    a^  =  iV  =  a,    whence,    log.  a  =  1  ; 
2(1  "  a"  zz:  1,  whence,    log.  1  =  0  ; 

that  is,  whatever  be  the  base  of  the  system,  its  logarithm  taken,  in 
that  system,   is  equal  to  1,  and   the  logarithm  of  1    is  equal   to   0 

264.  Let  us  suppose,  in  the  equation 

a'  zi:  N,  . 
that  •  a  >  1 . 

Then,  if  we  make   iV=  1,  we  shall  have 

aO=  1. 
If  we  make  iV<  1,  we  must  have 

a-'  =  N,     or     —  =  iV  <  1 . 
a* 

If  now,  N  diminishes,  x  will  increase,  and  when  N  becomes  0, 
we  have 

a-*  =  —  =  0,     or     c^  =  00  (Art.  Ill); 

but  no  finite  power  of  a  is  infinite,  hence  a;  =  oo  :  and  therefore, 
the  logarithm  of  0  in  a  system  of  which  the  base  is  greater  than 
vnity,  is  an  infinite  number  and  negative. 

265.  Again,  take  the  equation 

a==  =  N, 
and  suppose  the   base  a  <  1.     Then  making,  as  before, 

N  =z  1,   we  have   a°  =^  1. 


CHAP.  IX.]  LOGARITHMIC    SERIES.  279 

If  we  make  N  less  than  1,  we  shall  have 

a^  —  N  <:  1 . 

Now,  if  we  diminish  N,  x  will  increase;  for,  since  a  <  1,  its 
powers  will  diminish  as  the  exponent  x  increases,  and  when 
A^  =  0,  X  must  be  infinite,  for  no  finite  power  of  a  fraction  can 
be  0.  Hence,  the  logarithm  of  0  in  a  system  of  which  the  basf 
is  less  than  unity,  is  an  infinite  number  and  positive 

Logarithmic  and  Exponential  Series. 
266.  The  method  of  resolving  the  equation 

a-"  =  b, 

explained  in  Art.  255,  gives  an  idea  of  the  construction  of  log- 
arithmic tables ;  but  this  method  is  laborious  when  it  is  necessary 
to  approximate  very  near  the  value  of  x.  Analysts  have  discovered 
much  more  expeditious  methods  for  constructing  new  tables,  or  for 
verifying  those  already  calculated.  These  methods  consist  in  the 
development  of  logarithms  into  series. 
Taking  again  the  equation 

c-  =y, 

it  is  proposed  to  develop  the  logarithm  of  y  into  a  series  involving 
the  powers  of  y,  and  co-eflicients  independent  of  y. 

It  is  evident,  that  the  same  number  y  will  have  a  different  log- 
arithm in  ditferent  systems,  that  is,  for  different  values  of  the  base 
a;  hence,  the  log.  y,  will  depend  for  its  value,  1st,  on  the  value 
of  y ;  and  2dly,  on  a,  the  base  of  the  system  of  logarithms. 
Hence,  the  development  must  contain  y,  or  some  quantity  depen- 
dent on  it,  and  some  quantity  dependent  on  the  base  a. 

To  find  the  form  of  tl.is  development,  Ave  will  assume 

log.  y  =  A  +  By  +  Cy^  -\-  Dy3  -{-,  &c., 

m  which  A,  B,  C,  &c.,  are  independent  of  y,  and  dependent  on 
the  base  a. 

Now,  if  we  make  y  =  0,  the  log.  y  becomes  infinite,  and  is 
either  negative  or  positive,  according  as  the  base  a  is  greater  or 
less  than  unity  (Arts.  264  &  265).  But  the  second  member  un- 
der this  supposition,  reduces  to  A,  a  finite  number :  hence,  the 
development  cannot  be  made  imder  that  form 


280  ELEMENTS    OF    ALGEBRA.  [CHAP.     X. 

Again,  assume 

log.  y  =  Ay  +  By"^  +  Cy^  +  Dy^  +,  &c. 

If  we  make  y  =  0,  we  have 

log.  y  =  ±  OD,    that  is,     =b  oo  =  0, 
which  is  aljsurd,  and  hence  the  development  cannot  be  made  un- 
der the  last    form.     Hence   we    conclude   that,  the    logarithm   of  a 
number  catinot  be  developed  in  the  j)owers  of  that  number. 

Let  us  place,  in  the  first  member,   1  +  y    for  y,  and  we  have 
log.  (1  +  y)  =  Ay  +  By^  +  Cy^  +  Dy*  +  &c.  .  .  .  (1), 
making    y  =  0,  the  equation  is  reduced  to   log.  1=0,  which  does 
iiot  present  any  absurdity. 

In  order  to  determine  the  co-efficients  A,  B,  C,  .  .  .  we  shall 
follow  the  process  of  Art.  243.  Since  equation  (1)  is  true  for 
any  value  of  y,  it  will  be  true  if  we  substitute  z  for  y,  and  we 
may  write 

lug.  (1  +z)  =  Az  +  Bz^  +  Cz^  +  Dz*  +   .  .  .  (2). 

Subtracting  equation  (2)  from  (1),  we  obtain 

\og.{l^\-y)-\og.{\+z)=A{y-z)i-B{y^-z^~)+C{y^-z^)-{-..{3). 

The  second  member  of  this  equation  is  divisible  by  y  —  z.  Let 
us  see,  if  we  can  by  any  artifice,  put  the  first  under  such  a  form 
that  it  shall  also  be  divisible  by  y  —  z.     We  have, 

log.  (1  +  y )  -  log.  (1  +  ^)  =  log.  ( |-|-f )  =  log-  (i  +  f^)- 

But  since    — can    be    regarded    as    a    single    number   u,  we 

\  +  z 

can  develop  \og.[\-\-u),  or  log.  (l  +  ^— \   in  t-l^e  same  man- 
ner as  log.  (1  +  y),   which  gives 

Substituting  this  development  for 

log.(l+y)-log.(i+^), 

in    the  equation  (3),  and    dividing  both  members  by  y  —  z,  it  be- 
comes 

\  -{-  z         [l  -^  zf  (1  -r  zy 

=  A  +  B{y  ^  z)  +  C  {y-^-  +  yz  +  z-^)  +   .   .  . 


CHAP.  IX.] 


LOGARITII.AIIC    SERIES. 


281 


Since  this  equation,  like  the  preceding,  is  true  for  all  A^alues  of 
y  and  z,  make   y  =  ^,  and  there  will  result 

— —  =  ^  +  iBy  +  3Cy2  +  xBy'^  +  5£y*  +   .  .  . 
1  +y 

whence,  by  making  the  terms  entire,  and  transposing, 


y  +  3C 
+  2Z? 


y2  4-  4Z) 
+  3C 


+  M) 


y*  + 


.=.,  B=-4.  c=--=^-4.  p 


(  +  J.  +  2S 

Placing  the  co-efficients  of  the  different  powers  of  y  equal  to 
zero,  we  obtain  the  series  of  equations 

^_A  =  0,     25  +  A=0,     3C  +  2S=0,     4D4-3C==0 

whence, 

3C  _       ^ 

~~\  ~  ~T" 

The  law  of  the  series  is  evident ;  the  co-efficient  or"  the  n'^  term 

is  equal  to    ^:  — ,   according  as  n  is  even  or  odd  :  hence,  we  ob- 
n 

tain  for  the  development, 

AAA 
log.  (1  +y)  =vly-— y2  +  -_y3__y4  .  .  . 

Hence,  although  the  logarithm  of  a  number  cannot  be  developed 
in  the  powers  of  tliat  number,  yet  it  may  be  developed  in  the  powers 
of  a  number  less  by  unity. 

By  the  above  method  of  development,  the  co-efficients  B,  C,  D, 
E,  &c.,  have  all  been  determined  in  functions  of  A  ;  but  A  re- 
mains entirely  undetermined.  This  indeed  should  be  so,  since  A 
depends  for  its  value  on  the  base  of  the  system,  to  which  any 
value  may  be  assigned. 

Denote  by  a/  that  part  of  the  second  member  of  equation  (4) 
which  involves  y,  and  suppose  a  to  be  the  base  of  the  system  in 
which  the  log.  (1  +  y)   is  taken,  and  we  have 

a^^'=14-y,     or     ^a/ =  log.  (1  +  y). 

But  the  log.  (1  +  y)  depends  for  its  value  on  two  things  :  \iz., 
on  the  number  of  units  in  y,  and  on  the  base  of  the  system  in 
which  the  logarithm  is  taken.     The   series   denoted  by  x'  is   ex- 


282  ELEMENTS    OF    ALGEBRA.  [CHAP.   IX 

pressed  in  y,  and  hence  depends  for  its  value  on  y  alone.  Bu 
A  being  independent  of  y,  its  value  must  depend  on  the  base  of 
the  system ;    and  hence, 

The  expression  for  the  logarithm  of  any  number  is  composed  of 
tvo  factors,  one  dependent  on  the  number,  and  the  other  on  the  base 
cf  the  system  in  which  the  logarithm  is  taken.  The  factor  which 
depends  on  the  base,  is  called  the  modulus  of  the  system  of  log- 
arithms. 

267.  If  we  take  the  logarithm  of  1  +  y  in  a  new  system,  and 
denote  it  by  F.  (1  -f  y),  we  shall  have 

r.(l+y)  =  A'(y-|  +  ^-|  +  -|^-&c.)    (5), 

in  which  A'  is  the  modulus  of  the  new  system. 

If  we  suppose  y  to  have  the  same  value  as  in  equation  (4),  we 
shall  have 

V.{\  +y)  :  1.(1 +  y)  :   :  A'  :  A; 
for,  since  the  series   in   the   second   members   are   the  same,  they 
may  be  omitted.     Therefore, 

The  logarilh7ns  of  the  same  number,  taken  in  two  different  systems, 
are  to  each  other  as  the  7noduli  of  those  systems. 

268.  Having  shown  that  the  modulus  and  base  of  a  system  of 
logarithms  are  mutually  dependent  on  each  other,  it  follows,  thai 
if  a  value  be  assigned  to  one  of  them,  the  corresponding  value  of 
the  other  must  be  determined  from  it. 

If  then,  we  make  the   modulus 

A^=\, 
the  base   of  the   system  will  assume   a   fixed  value.     The  system 
of  logarithms  resulting  from  such  a  modulus,  — d  siich  a  base,  is 
called   the  Naperian  System.     This  was    the  first   system  known, 
and  was  invented  by  Baron  Napier,  a  Scotch  mathematician. 

With  this   modification,  the  proportion  above  becomes 
F.(l+y)  :  1.(1  +y)  :   :   I   :  A, 
and  .4.1'.(1  +y)  =  1.(1 +y). 

Hence  we  see  that. 

The  Naperian  logarithm  of  any  number,  multiplied  by  the  modu- 
lus of  another  system,  will  give  the  logarithm  of  the  same  number 
in  that  system. 


CHAP.  IX.]  LOGARITHMIC    SERIES.  283 

The  modulus  of  the  Naperian  System  being  unity,  it  is  found 
most  convenient  to  compare  all  other  systems  with  the  Naperian  ; 
and  hence,  the   modulus  of  any  system  may  be   defined  to   be, 

The  number  by  which  it  is  necessary  to  multiply  the  Naperian 
logarithm  in  order  to  obtain  the  logaritJnn  of  the  same  number  in 
the   other  system. 

269.  Again,         A  X  l'.(l  +  y)  =  1.(1  +  y)     gives 

■    .'.(!  + y)=!:<i±l). 

That  is,  the  logarithm  of  any  number  divided  by  the  modulus  of 
its  system,  is  equal  to  the  Naperian  logarithm   of  the  same  number. 

270.  If  we  take  the  Naperian  logarithm  and  make  y  =  1,  equa- 
tion (5)  becomes 

2^3         4^5 

a  series  which  does  not  converge  rapidly,  and  in  which  it  would 
be  necessary  to  take  a  great  number  of  terms  to  obtain  a  near  ap- 
proximation. In  general,  this  series  will  not  serve  for  determining 
the  logarithms  of  entire  numbers,  since  for  every  number  greater 
than  2  we  should  obtain  a  series  in  which  the  terms  would  go 
on  increasing  continually. 

The  following  are  the  principal  transformations  for  converting 
the  above  series  into  converging  series,  for  the  purpose  of  obtain- 
ing the  logarithms  of  entire  numbers,  which  are  the  only  logarithms 
placed  in  the  tables. 

First   Transformation. 

Taking  the  Naperian  logarithm  in  equation  (5),  making  y  =  — , 

sr 

'.nd  observdnff  that 


F.(l  +  1)=F.(1  +  ;^)_K^,    it 


becomes 


I'.(l+.)_l'..  =  l-^+^_-l^  +  &c  (6), 

Tliis  series  becomes  more  converging  as  z  increases  5  besides, 

tne  first  member  of  the  equation  expresses  the  difference  between 
the  logarithms  of  two  consecutive  numbers. 


8        24 

1 
-6i+-- 

18         81 

324  "^ 

1           1 

1 

1 

284  ELEMENTS    OF   ALGEBRA.  [CHAP.  IX. 

Making  2  =  1,  2,  3,  4,  5,  &c.,  in  succession,  we  have 

1  1  1  1  o 

V:2  —  1 1 \ &c. 

2  3         4         5 

r.3-r.2  =  — - 

2 

F.4-r.3  =  — - 
3 

F.5  -  VA  =~-         , 

4         32       192        1024 

The  first  series  will  give  the  logarithm  of  2  ;  the  second  series 
will  give  the  logarithm  of  3  by  means  of  the  logarithm  of  2 ;  the 
third,  the  logarithm  of  4,  in  functions  of  the  logarithm  of  3  .  .  .  &;c. 
The  degree  of  approximation  can  be  estimated,  since  the  series 
are  composed  of  terms  alternately  positive  and  negative  (Art.  245) 


Second   Transformation. 

A  much   more    converging   series    is    obtained   in   the    following 
manner.     In  the   series 

^  ^  2         3         4 

substitute   —  x  for  x,  and  it  becomes 


V.n-x)=  -X-  —  -  —  -  — 

^  ^  2  3         4 


Subtracting  the  second  series  from  the  first,  observing  that 

K(l  +  x)  -  V.n  -  x)  =  V.  f  ^^t^),     we  obtain 

W  ~  X  / 

(\  -\-  x\       n  /      ,    a;3        a?5        x"^        x'^  \ 

This    series  will  not    converge  very  rapidly  unless  a;  is  a  very 

I  -\-  X 
small  fraction,  in  which  case, will  be  greater  than  unity, 

but  will  differ  very  little  from  it. 

\  -{-  X  1 

Make    — =  1  H ,    z  being  an  entire  number.     We  have 

1  —  X  z 

(1  +  a-)  2  =  (1  —  x){z  ^  V);     whence,     a;  = 


1z  +  1 


CHAP.  IX.] 


LOGARITHMIC    SERIES. 


285 


Hence,  the  preceding  series  becomes    1'.  (l  H ),    or 

•'■(^ +  "-•'—  =  (iJTT  +  spITl?  +  512JTTF +••  ■)• 

This  series  gives  the  difference  between  the  logarithms  of  two 
consecutive  numbers,  and  converges  more  rapidly  than  series  (6) 
Making  successively,   z=l,  2,  3,  4,  5  .  .  .,    we  find 


''2  =  *(t+03  + 

l'.3-I'.2=2(-i+3ip  + 
l'.4-l'.3  =  3  (1+3^3  + 

/    1  1 


V.5  -  VA  =  2 


+ 


+ 


75 


+ 


+ 


95       7.9 


7.3^         ■  ■  ■/' 

7.5^  ^   ■  ■  7' 
-^+         ) 


Let    2:=  100;    there  will  result 


IMOl^ilMOO 


+  ^2^ 


+ 


1 


+ 


1 


+  ...); 


201     '    3(201)3       5(201)5 

whence  we  see,  that  knowing  the  logarithm  of  100,  the  first  term 
of  the  series  is  sufficient  for  obtaining  that  of  101  to  seven  places 
of  decimals. 

There  are  formulas  more  converging  than  the  above,  from  which 
we  may  obtain  a  series  of  logarithms  in  functions  of  others  al- 
ready known,  but  the  preceding  are  sufficient  to  give  an  idea  of 
the  facility  with  which  tables  may  be  constructed.  We  may  now 
suppose  the  Naperian  logarithms  of  all  numbers  to  be  known. 

The  Naperian  logarithm  of  10  may  be  deduced  from  the  first 
and  fourth  of  the  above  equations,  by  simply  adding  the  logarithm 
of  2  to  that  of  5  (Art.  258).  This  number  has  been  calculated 
with  great  exactness,  and  is  2.302585093. 

271.  We  have  already  observed  that  the  base  of  the  common 
system  of  logarithms  is  10  (Art.  257).  We  will  now  find  iis 
modulus.     We  have, 

r.(l  +  V)  :  1-(1  +  y)  :  :   1   :  ^     (Art.  267). 


286  ELEMENTS    OF    ALGEBRA.  [CHAP.  IX. 

If  we  make    y  =  9,    we  shall  have 

IMO  :  1.10  :  :  1  :  ^. 
But  the  IMO  =  2.302585093,      and     1.10  =  1   (Art.  257)  ; 

hence,   A  — —  =  0.434294482  =    the    modulus    of   the  . 

2.302585093 

common  system. 

If  now,  we  multiply  the  Naperian  logarithms  before  found,  by 
this  modulus,  we  shall  obtain  a  table  of  common  logarithms 
(Art.  268). 

All  that  now  remains  to  be  done  is  to  find  the  base  of  the  Na- 
perian system.  If  we  designate  that  base  by  e,  we  shall  have 
(Art.  267), 

\' .e  :  \.e  :   :   \   :  0.434294482. 

But  \'e~  1   (Art.  263):    hence, 

1.   :  l.e  :   :   1   :  0.434294482, 
hence,  l.e  =  0.434294482. 

But  as  we  have  already  explained  the  method  of  calculating 
the  common  tables,  we  may  use  them  to  find  the  number  whose 
logarithm  is  0.434294482,  which  we  shall  find  to  be  2.718281828: 
hence 

e  =  2.718281828. 

We  see  from  the  last  equation  but  one  that,  the  modulus  of  the 
common  system  is  equal  to  the  common  logarithm  of  the  Naperian  base 

Of  Interpolation. 

2T2.  A  table  of  logarithms  is  a  tabulated  series  of  numbers, 
showing  the  value  of  x  in  the  equation 

a'^  =  N, 
corresponding  to  all  the  integral  values  of  N,  between  1  and  some 
higher  number  which  marks  the  limit  of  the  table.     It  has  already 
been  remarked  that  in  the  system  in  common  use,  the  value  of  the 
base   a,   is  10. 

And  generally,  awy  mathematical  table  consists  of  a  series  of  values 
of  some  letter  in  an  algebraic  expression,  corresponding  to  equi-dis 
tant  values  of  the  function  on  which  it  depends. 

The   principle  of  interpolation,  which  is  of  great  value  in  prac 
tical  science,  has  for  its  object  to  find  from  the  tabulated  number* 


CHAP.   IX.] 


OF    IXTERPOLATION. 


287 


which  are  given,  other  similar  numbers  which  shall  correspond 
to  intermediate  values  of  the  function.  For  example,  suppose  p,  q, 
r,  s,  &c.,  to  be  a  series  of  tabulated  numbers  corresponding  to, 
and  written  opposite  the  functions  a,  a  -{-  b,  a  -\-2b,  o  +  3i,  &c., 
and  it  were  required  to  find  the  tabulated  number  corresponding 
to  the  function  a  +  2\h.  This  is  a  question  of  interpolation,  and 
is  resolved  by  taking  the  successive  differences  of  the  tabulated 
numbers,  thus : 

Functions.  Differences  of  tabulated  numbers 


a 

a  -\-  h 

a  +  2h 

a  +  U 

a  +  Ah 


p     -       .         - 

dp      - 

q  d?p        - 

dq  d^p 

r  d'^q 

dr  d^q 

s  d^r 

ds       - 
t 


in  which    dp  =  q  —  p,  dq  ^^  r  —  q,  dr  =  s  —  ■^^  &c  ; 

and  '^p  =  dq  —  dp,        d'^q  =  dr  —  dq,        d-r  =  ds  —  dr,  &c. ; 

also,  d^p  =  d'^q  —  d^p,     d^q  =  d'^r  —  d^q,  &c. 

&c.,  (fee. 

From  the   above   equations,  we  have 

q  =p  -\-  dp,     r  =  q  -}-  dq  =  p-j-  dp-\-  dq  =p  -\-  2dp  +  d^p  ; 
and  by  a  similar  process,  we  have 

s  =  p  +  3dp-\-  3 J2p  4-  d^p, 
t  =  p  -^  4dp  -\-  ed'^p  +  id^p  +  d*p, 
&c.,  &c., 

m  which  notation  it  should  be  observed,  that  d-,  d^,  &c.,  denote  the 
second,  third,  &,c.  differences  of  the  successive  tabulated  numbers 
It  is  plain,  that  the  above  law  from  which  the  numerical  co- 
efficient for  any  term  may  be  derived,  is  similar  to  tliat  for  the 
co-efficients  of  a  binomial:  hence,  if  T  denote  the  71+ 1  term  of 
the   tabulated  numbers,  reckoning  from  p  inclusive,  we  shall  have 

^  ,      ,    n(n  —  1)  ,„         n  .  (n  —  I)  (n  —  2)  „      ,     „ 

T  =  p  +  ndp  -\-  -^ ^  d^p  -i ^ ^-^ ^  d^p  +  &c. 

^"'1.2  •'^       I  .         2.3  ^ 


288  ELEMENTS    OF    ALGEBRA.  [CHAP.   IX. 

Let  it  be  required  to  find  the  tabulated  number  correspondiug 
to    a  +  3b.     We  then  have,    n  =  3  :    hence, 

T  =  p  +  3dp  +  3(Pp  +  d?p, 

the  same   value  as   that  found  above  for  s. 

Next,  let  it  be  required  to  find  the  tabulated  value  answeriii;! 
to  the  functions  a  -\-  ^b.  Then,  w  =  f ,  and  if  we  know  the  tab- 
ulated number  p,  and  the  successive  differences  d,  d^,  &c.;  tlic 
approximate   value   of  T  can   easily  be   found. 

It  is  plain  from  the  series  that  the  interpolated  values  are  bul 
approximations,  since  no  order  of  difference  can  reduce  to  zero , 
and  hence,  the  series  will  contain  an  infinite  number  of  terms. 
Generally,  however,  the  tabulated  values  are  themselves  but  ap- 
proximations, and  the  successive  differences  decrease  so  rapidly 
in  value,  that  the   series   becomes   very   converging. 

Let  us  suppose  for  example,  that  we  have  the  logarithms  of 
12,  13,  14,  15,  &c.,  and  that  it  is  required  to  find  the  logarithnr. 
of  12  and  a  half.     Then, 

12  I    1.079181 

I  dp  z=  0.034762 

13  1.113943  (Z2p  =— 0.002577 

dq  =  0.032185 

14  1.146128  d^q  zzz  —  0.002222 

dr  =  0.0299G3 

15  1.176091 

also,  d'^p  =  d^-q  —  d^p  =  +  0.000355. 

Making  n  =  — ,  and  stopping  at  the  term  involving  the  third 
difference,  we  have 

T^p  +  ^dp-^^X-^d^p+—X-—X-Y(^'P  +  ^'^ 

+         p =  1.079181 

-\-     ^dp         -         '         '         -         -         =  0.017381 

-f    Id'^p =  0.000322 

+  ^\d3p =  0.000022 

T  =  log.  121  =  1 .090906. 


CHAP.  IX.]  FORMULAS  FOR  INTEREST.  28& 


INTEREST. 

273.  The  solution  of  all  questions  relating  to  interest,  may  be 
greatly  simplified   by  employing  the  algebraic   formulas. 

In  treating  of  this  subject,  we  shall  employ  the  following  no 
tation  : 

Let   p  =    the  amount  bearing  interest,  called  the  principal ; 

r  =    the  part  of  $1,  which  expresses   its  interest  for  one 
year,  called  the  rate  per  cent. ; 

t  =  the  time,  in  years,  that  2^  draws  interest ; 
i  =  the  interest  of  2^  dollars  for  t  years ; 
S  ==■  p  -\-  the  interest  Avhich  accrues  in  the  time  t.     This 
surn  is  called  the  amou7Xt. 

Simple  Interest. 

To  jind  the  interest  of  a  sum  p  for  t  years,  at  the  rate  r,  and 
the  amount  then  due. 

Since  r  denotes  the  part  of  a  dollar  which  expresses  its  in- 
terest for  a  single  year,  the  interest  of  p  dollars  for  the  same  time 
will  be  expressed  by  pr ;  and  for  t  years  it  will  be  t  times  as 
much :    hence, 

i=ptr (1); 

and  for  the  amount  due, 

S  =z  p -]- ptr  =  p  {}  -\-  tr)  .  .  (2). 

EXAMPLES. 

1.  What  is  the  interest,  and  what  the  amount  of  $365  for  three 
years  and  a  half,  at  the  rate  of  4  per  cent,  per  annum.     Here, 

p  =  $365; 

4 

r  = =  0.04  ; 

100 

«  =  3.5; 

i=ptr=  365  X  3.5  X  0.04  r=  $51,10: 

hence,  -S  =  365  +  51,10  =  $416,10. 

19 


'290  ELEMENTS    OF    ALGEBRA.  [CHAP.   IX. 

Present    Value  and  Discount  at  Simjyle  Interest. 

The  present  value  of  any  sum  «S,  due  t  years  hence,  is  the  prin- 
cipal p,  which  put  at  interest  for  the  time  t,  will  produce  the 
amount   S. 

The  discount  on  any  sum  due  t  years  hence,  is  the  difference 
between  that  sum  and  the  present  value. 

To  find  the  present  value  of  a  sum  of  dollars  denoted  hy  S,  due 
t,  years  hence,  at  simple  interest,  at  the  rate  r  ;   also,   the  discount 

We  have,  from  formula  (2), 

S  —p  +  ptr; 
and  since  p  is  the    principal  which  in   t   years   will   produce   tho 
sum  S,  we  have 

p  =  ~  ■■■(?)■' 

and  for  the  discount,  which  we  will  denote  by  Z),  we  have 

D  =  S ^  =  -^...(4). 

1  +  «r         1  +  ir  ^  ^ 

1.  Required  the   discount  on  $100,  due  3  months  hence,  at  the 

rate   of  5^    per   cent,  per   annum. 

S=SiOO  =1  $100 

t  =  3  months  =  0.25 

5.5 


100 
Hence,  the  present  value  p  is 

S  100 


=  .055. 


$98,643 


^        I  -\-tr        1  +  .01375 
hence,  D  =  >S  -  p  =  100  —  98,643  =  $1,35  7. 

Compound  Interest. 

Compound  interest  is  when  the  interest  on  a  sum  of  money  be- 
coming due,  and  not  paid,  is  added  to  the  principal,  and  the  in- 
terest then  calculated  on  this  amount  as   on   a  new  principal. 

To  find  the  amount  of  a  sum  p  placed  at  interest  for  t  yean, 
compound  interest   being  allowed  annually  at   the   rate  r. 

At  the   end  of  one   year  the   amount  will  be 
S  =p  +pr  =  p{\  -{-  r). 


CHAP.   IX.]  FORMULAS    FOR    INTEREST.  L'l'l 

Since  compound  interest  is  allowed,  this  sum  now  becomes  the 

principal,  and  hence   at  the   end   of  the   second  year  the   amount 

will  be 

S'=p(l  +  r)+pr{l  +r)=p(l  +  r)2. 

Regard  /» (1  +  r)^  as  a  new  principal;  we  have,  at  the  end  of 

the  third  year, 

S''=p(l  +r)2+pr(l-hr)2=p(l  +  r)3 ; 

and  at  the  end  of  t  years, 

S=p{l-\-rY (5). 

And  from  Article     2G0    we  have 

\og.S  =  \og.p  +  t\og.(l  +r); 

and  if  any  three  of  the  four  quantities  S,  p,  t,  and  r,  are  given, 
the  remaining  one  can  be  determined. 

Let  it  be  required  to  find  the  time  in  which  a  sum  p  will  double 
itself  at  compound  interest,  the  rate  being  4  per  cent,  per  annum. 

We  have  from  equation  (5), 

S=p{\+rY. 

But  by  the  conditions  of  the  question, 

S  =  2p  =  p{\-\-ry: 

hence,  2  =  (1  +  r)', 

_       log.2        _  0.301030 
^^  ^  ~  log.  (1  +  r)  ~  0.017033 

=  17.673  years 

=  17  years,  8  months,  2  days. 

To  find  the  Discount. 

The  discount  being  the  difference  between  the  sum  (S  and  p, 
we  have 


292  ELEMENTS  CF  ALGEBRA  [CHAP.  X. 


CHAPTER  X. 

t 

GENERAL  THEORY  OF  EQUATIONS. 

274.  The  most  celebrated  analysis  have  tried  to  resolve  equa- 
tions of  any  degree  whatever,  but  hitherto  tlieir  efforts  have  been 
unsuccessful  with  respect  to  equations  of  a  higher  degree  than  the 
fourth.  However,  their  investigations  have  conducted  them  to  some 
properties  common  to  equations  of  every  degree,  which  they  have 
since  used,  either  to  resolve  certain  classes  of  equations,  or  to  re 
duce  the  resolution  of  a  given  equation  to  that  of  one  more  simple 
In  this  chapter  it  is  proposed  to  make  known  these  properties, 
and  their  use  in  facilitating  the  resolution  of  equations. 

The  development  of  the  properties  of  equations  of  any  degree, 
leads  to  the  consideration  of  polynomials  of  a  particular  nature, 
and  entirely  different  from  those  considered  in  the  first  chapter. 
These  are  expressions  of  tlie  form 

Ax""  +  Bx'"-'  +  Ca:'»-2  -\-   .  .   .    ^  Tx-\-  U, 

in  which  m  is  a  positive  whole  number ;  but  the  co-efficients 
A,  B,  C,  .  .  .  T,  U,  any  quantities  whatever,  that  is,  entire  or 
fractional,  commensurable  or  incommensurable.  Now,  in  algebraic 
division,  as  explained  in  Chapter  II.,  the  object  was  this,  viz.  : 
having  given  two  polynomials,  entire  with  reference  to  all  the  let- 
ters and  particular  numbers  involved  in  them,  to  find  a  third  poly- 
ncmial  of  the  same  kind,  ichich  multiplied  by  the  second  shall  jjro- 
duce  the  first. 

But  when  we  have  two  polynomials, 

Ax^  +  Ex-"-!  +  Ca;"'-2  -\.   .   ,   .    ^  Tx  -\-  U, 
A'a^  -\-  5^r"'-i   I-  C'.T"'-2  Jf.    .   .   .   J^  Tx  ^  U, 
which  are  necessarily  entire  only  with  respect  to  x,  and  in  which 
the    cO'Cfficients    A,   B,    C  .  .  .,   A',  B' ,   C  .  .  .,  are    any  quan- 
tises whatever,  it  may  be  proposed  to  find  a  third  polynomial,  of 


CHAP.  X.]         GENERAL  THEORY  OF  EQUATIONS.  293 

the  same  form  and  nature  as  those  that  are  given,  which  multiplied 
bij  the  second  vnll  re-produce  the  first. 

275.  Ordinary  polynomials,  that  is,  polynomials  which  are  en- 
tire with  reference  to  all  the  exponents  and  co-efficients,  are  called 
rational  and  entire  polynomials.  Polynomials  which  are  only  en- 
lire  with  reference  to  the  letter  x,  and  whose  co-efficients  are  any 
(iuantities  whatever,  are  called  entire  functions  of  x. 

276.  Every  complete  equation  of  the  rn"'  degree,  m  being  a  pos- 
itive whole  number,  may,  by  the  transposition  of  terms,  and  by 
the  division  of  both  members  by  the  co-efficient  of  x",  be  put  un- 
der the  form 

a;'"  +  Px'"-!  +  Qx'"-2  -{-.,,    ^  Tx  +  U  =0, 

P,  Q,  R  .  .  .,  T,  U,  being  co-efficients  taken  in  the  most  general 
algebraic  sense. 

Ani/  expression,  which  substituted  in  place  of  x  satisfies  the  equa- 
tion, that  is,  renders  its  first  member  equal  to  0,  is  called  a  root  of 
the  equation. 

111.  As  every  equation  may  be  considered  as  the  translation 
into  algebraic  language  of  the  relations  which  exist  between  the 
given  and  unknown  quantities  of  a  problem,  we  are  naturally  led 
to  suppose  that,  every  equation  has  at  least  one  root.  We  will 
admit  this  principle,  which  we  shall  have  occasion  to  verify  here- 
after for  most  equations. 

We  will  now  demonstrate  some  of  the  principal  properties  of 
a  general  equation. 

First  Property. 

278.  In  every  general   equation  of  the  form 

a;m  ^  Px^-^  +  Qj''"-2  +   .  .  .   J^Tx-\-  TJ  =Q, 

the  first  member  is  divisible  by  the  difference  between  the  un- 
known quantity  x  and  a  root  of  the  equation ;    that  is, 

7/'  a  is  a  root  of  the  equation,  the  first  member  will  be  exactly 
divisible  by  x  —  a  ;  and  reciprocally,  if  a  divisor  of  the  form  x  —  a 
will  exactly  divide  the  first  member,  a  will  be  a  root  of  the  equation. 

Let  us  suppose  the  first  member  of  the  proposed  equation  to  be 
divided  by   x  —  a,   and  the  operation  continued  until  all  the  terms 


294  ELEMENTS  OF  ALGEBRA  [CHAP.  X 

involving  x  are  exhausted :    the    remainder,  if  there    be    any,  will 
then  be  independent  of  x. 

If  we  represent  the  remainder  by  R,  and  the  quotient  obtained 
y   Q',  we  may  write 

a:'"  +  Px^-^  ....    ^^  Tx  -\-  TJ  =  Q'  {x  -  a)  -\-  R. 

Now,  since  by  hypothesis,  a  is  a  root  of  the  equation,  if  we 
substitute  a  for  x,  the  first  member  of  the  equation  will  reduce  to 
zero;  the  term  Q' {x.  —  a)  will  also  reduce  to  0,  and  consequently, 
we  shall  have 

R=0. 

But  since  R  does  not  contain  x,  its  value  will  not  be  affected 
by  attributing  to  x  the  particular  value  a :  hence,  the  remainder  R 
was  originally  equal  to  zero,  and  consequently,  the  first  member 
of  the  equation 

^m  _|.  p^m-l  _|_    Q^m-2   ,   .   ,   .    -\.    Tx  +   U  —  0, 

is  exactly  divisible  by  x  —  a. 

Reciprocally,  if  a;  —  a  is  an  exact  divisor  of  the  first  member 
of  the  equation,  the  quotient  Q'  will  be  exact,  and  we  sluill  have 
R  —  0:    hence, 

jnm  _j_  p^m-\    .    .    ,    ^    Tx  +    U  =   Q'  {x  —  a). 

If  now,  we  suppose  x  =  a,  the  second  member  will  reduce  to 
zero,  consequently,  the  first  will  reduce  to  zero,  and  hence  a  will 
be  a  root  of  the  equation  (Art.  276).  It  is  evident,  from  the  na- 
ture of  division,  that  the  quotient    Q'  will   be   of  the   form 

^m-l  _f.  p'xm-2 _^  7^/  +    C//  _  0. 

279.  It  follows  from  what  has  preceded,  that  in  order  to  dis- 
cover whether  any  polynomial  is  exactly  divisible  by  the  binomial 
X  —  a,  it  is  sufficient  to  see  if  the  substitution  of  a  for  a;  will 
reduce  the  polynomial  to  zero. 

Reciprocally,  if  any  polynomial  is  exactly  divisible  by  x  —  a, 
then  we  know,  that  if  the  polynomial  Tje  placed  equal  to  zero,  a 
will  be  a  root  of  the  equation. 

The  property  which  we  have  demonstrated  above,  enables  us 
to  diminish  the  degree  of  an  equation  by  unity  when  we  know 
one  of  its  roots,  by  a  simple  division ;  and  if  two  or  more  of  the 
roots  are  known,  the  degree  of  the  equation  may  be  still  further 
diminished  by  continuing  the   division. 


CHAP.  X.]         GENERAL  THEORY  OF  EQUATIONS.  295 

EXAMPLES. 

1.  One  of  the  roots  of  the  equation 

X*  —  25a;2  +  60a:  —  36  =  0 
is  3 :    what  is  the  equation  containing  the  other  roots  ? 
X*  -  25x2  -(-  60x  -  36  ||a;   -  3 
X*  —    3^3  a;3  +  3a:2  -1674^12 

+    3x3  _  25x2 
3x3  _     9^2 


—  1 6x2  +  60x 

—  16x2  +  48x 


12x  —  36 
12x—  36 
Ans.  x^  -f  3x2  —  ]6x  +  12  =  0. 

2.  Two  roots   of  the  equation 

X*  —.12x3  _^  48a;2  _  68x  -f  15  =  0 

are  3  and  5 :    what  is   the   equation   containing  the   other   two  ? 

Ans.  x2  —  4x  -(-  1  =  0. 

3.  One   of  the   roots   of  the   equation 

x3  —  6x2  +  llx  — 6  =  0 
is  1  :    what  is  the  equation  containing  the  other  roots  ? 

Ans.  x^  —  5x  +  6  =  0 

4.  Two  of  the  roots  of  the   equation 

4x*  —  14x3  —  5x2  -f-  31x  +  6  =  0 
are  2   and   3  :    find  the   equation  containing  the   other  roots. 

Ans.  4x2  _f_  5a;  _j_  1  -_  0. 

Second  Property. 

280.  Every  equation  involving  but  one  unknown  quantity,  has  as 
many  roots  as  there  are  units  in  the  exponent  which  denotes  its  de- 
gree,  and  no   more. 

Let  the  proposed  equation  be 

x'»  +  Px'"-!  +  Qx'"-2  +  .  .  .  4-  Tx  +  C/  =  0. 

Since  every  equation  is  supposed  to  have  at  least  one  root  (Art. 
277),  if  we  denote  that  root  by  a,  the  first  member  will  be  divisi- 
ble by   X  —  a,    and  we   shall  have  the   equation 

x«  +  Pa-""-^  -}-...  ={x  —  a)  (x"*-!  +  P^x^-2  +...)•••  (H 


296  ELEMENTS    OF    ALGEBRA.  [CHAP.  X 

But  if  we  place 

we  obtain  an   equation  which  has   at  least  one  root. 
Denote   this   root  by  b,   we   have  (Art.  278), 

a™-i  +  P'x^-"^  -\-  .  .  .  =.{x  ~b)  (x'"-2  +  P''x^-^  +...). 

Substituting  the    second  member,  for   its  value    in  equation  (11 
and  we  have, 

^m  _^  Pa;m-i  J^  _,=,{^x  —  a){x  —  h)  (a;'"-2  +  P''  a;'"-3  +...)..  (2) . 

Reasoning  upon  the  polynomial 

as  upon  the   preceding  polynomial,  we  have 

^m-2  ^  P'^x^-^  +  .  .  .  =  (a;  —  c)  (a;'"-3  +  P"'oi?^-*-  +...), 

and  by  substitution,  , 

x'«  4-  Px'^"^  +  .  .  .  =  {x—  a){x  —  b){x  —  c)  (a''"-^  +  P^'x'^-^)     (3) 

281.  Observe,  that  for  each  binomial  factor  of  the  first  degree  with 
reference  to  x,  the  degree  of  x  in  the  polynomial  is  diminished 
by  unity ;  therefore,  after  m  —  2  factors  of  the  first  degree  have 
been  divided  out,  the  exponent  of  x  will  be  reduced  to  m  —  (m  —  2) 
=  2  ;  that  is,  we  shall  obtain  a  polynomial  of  the  second  degree 
with  reference  to  x,  which  can  be  decomposed  into  two  factors 
of  the  first  degree  (Art.  142),  of  the  form  x  —  k,  x  —  I.  Now, 
supposing  the  m  —  2  factors  of  the  first  degree  to  have  already 
been  indicated,  we   shall  have   the   identical   equation, 

x'^-i-  Pa;'"-i+  .  .  .  =(x  ~  a){x  —  b){x  —  c)  .  .  .{x  —  k)  (x  —  I)  zzzO; 

from  which  we  see,  that  the  Jirst  member  of  the  proposed  equation 
may  be   decomposed  into  m   binomial  factors  of  the  first   degree. 

As  there  is  a  root  corresponding  to  each  binomial  divisor  of 
the  first  degree  (Art.  278),  it  follows  that  the  m  binomial  factors 
of  the  first  degree,  x  —  a,  x  —  b,  x  —  c  .  .  .,  give  the  m  roots, 
a,  b,  c  .  .  .,  of  the   proposed  equation. 

But  the  equation  can  have  no  other  roots  than  a,  b,  c  .  .  .  k,  I 
For,  if  it  had  a  root  a'',  different  from  a,  b,  c  .  .  .  I,  it  would  have 
a  divisor  x  —  a'',  different  from  x  —  a,  x  —  b,  x  —  c  .  .  .  x  ■  I, 
which  is  impossible.     Therefore,  finally, 


CHAP.  X.]  COMPOSITION    OF    EQUATIONS.  297 

Every  equation  of  the  \x\^^  degree  has  m  roots,  and  can  have  no 
more. 

282.  In  equations  which  arise  from  the  multiplication  of  equal 
factors,  such  as 

{x  -  ay  (x  —  by  (x  —  cy  {x  —  d)  =  0, 

ihe  number  of  roots  is  apparently  less  than  the  number  of  units  in 
the  exponent  which  denotes  the  degree  of  the  equation.  But  this 
is  not  really  so ;  for,  the  above  equation  actually  has  ten  roots, 
four  of  which  are  equal  to  a,  three  to  b,  two  to  c,  and  one  to  d. 
It  is  evident  that  no  quantity  a\  different  from  a,  b,  c,  d,  can 
verify  the  equation  ;  for,  if  it  had  a  root  a',  the  first  member 
would  be  divisible  by   a?  —  a',  which  is  impossible. 

Consequence  of  the  second  Property. 

283.  It  has  been  shown  that  the  first  member  of  every  equation 
of  the  m^^  degree,  has  m  binomial  divisors  of  the  first  degree,  of 
the  form 

X  —  a,     X  —  b,     X  —  c,  .  .  .  X  —  k,     X  —  I. 

If  we  multiply  these  divisors  together,  two  and  two,  three  and 
three,  &c.,  we  shall  obtain  as  many  divisors  of  the  second,  third, 
&,c.  degree,  with  reference  to  x,  as  we  can  form  different  com- 
binations of  m  quantities,  taken  two  and  two,  three  and  three,  &c. 
Now  the  number  of  these  combinations  is   expressed  by 


m  —  1  m  —  1       m  —  2 

2      '  2  3 

Hence,  the  proposed  equation  has 

m  —  1 


(Art.  201). 


2 
divisors  of  the  second  degree ; 

rn  —  1       m  —  2 


2 
devisors  of  the  third  degree  ; 

m  —  1       m  —  2 


2  3  4 

divisors  of  the  fourth  degree  ;    and  so  on 


298 


ELEMENTS    OF    ALGEBRA. 


[CHAP.  X 


Composition  of  Eqi/atio7is. 
284.  If  in  the  identical  equation 

a:"*  +  Px'"-^  +  ...  =  (x  —  a)  {x  —  b)  {x  —  c)  .  .  .  {x  —  I), 

•we  perform  the  multiplication  of  four  factors  in  the  second  meiii- 
ber,  we  have, 

—  abc 

—  abd 

—  acd 

—  bed 


—  a 

x^  +  ab 

~b 

+  ac 

—  c 

+  ad 

-d 

+  be 

+  bd 

+  ed 

X  -j-  abed  ~ 


V  =0. 


If  we  perform  the  multiplication  of  the  m  factors  of  the  second 
member,  and   compare   the   terms   of  the    two   members,   we    shall 
find  the  following  relations  between  the  co-efficients  P,  Q,  R,  .  . 
T,  U,  and  the  roots  a,  b,  e,  .  .  .  k,  I,  of  the  proposed  equation,  viz.. 

^a  —  b  —  c...—k  —  l  =  P,    or    a-{- b +e  +  .  .  .  +  k+ I  =  -  P  ; 

ab  +  ae  +  .  .  .    -\-  kl  =  Q, 

—  abc  —  abd  ...    —  ikl  ^^  R,    or    ahe  -\-  abd -\-  ikl  =  —  R  ; 


dt  abed  .  .  ,  kl  =  U,    or    abed  .  .  .  kl  =  ±  U. 

The  double  sign  has  been  placed  in  the  last  relation,  because 
the  product  —  a  x  —  b  x  —  c...  x  —  I  will  be  plus  or  minus 
according  as   the   degree   of  the  equation  is   even  or  odd.     Hence, 

1st.  The  algebraic  sum  of  the  roots,  taken  with  contrary  signs, 
is  equal  to  the  co-efficient  of  the  second  term  ;  or,  the  algebraic 
sum  of  the  roots  themselves,  is  equal  to  the  co-efficient  of  the 
second  term  taken  with  a  contrary  sign. 

2d.  The  sum  of  the  products  of  the  roots  taken  two  and  two, 
with  their  respective  signs,  is  equal  to  the  co-efficient  of  the  third 
term. 

3d.  The  sum  of  the  products  of  the  roots  taken  three  and  three, 
with  their  signs  changed,  is  equal  to  the  co-efficient  of  the  fourth 
term ;  or  the  co-efficient  of  the  fourth  term,  taken  with  a  contrary 
sign,  is  equal  to  the  sum  of  the  products  of  the  roots  tjiken  three 
and  three ;    and  so  on. 


CHAP.   X.]  COMPOSITION    OF    EQUATIONS.  299 

4th.  The  product,  of  all  the  roots,  is  equal  to  the  last  term  ; 
that  is,  the  product  of  all  the  roots,  taken  with  their  respective 
signs,  is  equal  to  the  last  term  of  the  equation,  taken  with  its  sign, 
when  the  equation  is  of  an  even  degree,  and  with  a  contrary  sign, 
when  the  equation  is  of  an  odd  degree.  If  one  of  the  roots  is  equal 
to  0,  the  absolute  term  will  he  0. 

The  properties  demonstrated   (Art.    143),   with   respect   to  equa 
tions  of  the  second  degree,  are  only  particular  cases  of  the  above 

Consequences. 

1.  If  the  co-efficient  of  the  second  term  of  an  equation  is  equal 
to  zero,  the  term  will  not  appear  in  the  equation  ;  and  the  sum 
of  the   positive  roots  is   equal  to  the  sum  of  the  negative  roots. 

2.  Every  commensurable  root  of  an  equation  is  a  divisor  of  the 
last  or  absolute  term. 

EXAMPLES    IN'    THE    FORMATION    OF    EQUATIONS. 

1.  Form   the  equation  whose  roots   are   2,   3,  5,  and    —  6. 
We  have,  by  simply  indicating  the  multiplication  of  the  factors, 

(x  —  2){x  —  3)  {x  —  5)  (a;  +  6)  =  0. 

But  the  process  may  be  shortened  by  detaching  the  co-efficients 
thus : 

1—    2   |-3 
—    3  +~& 


1  — 

5+    6   (-5 
5  +  25  —  30 

1  - 

10  +  31-30   1+6 
6  -  60  +  186  -  180 

1  - 

4  —  29  +  156  —  ISO. 

Hence,  the   required  equation  is 

x^  -  4x3  _  29x2  _|.  156a;  _  180  =  0. 

2.  What  is   the  equation  whose   roots   are    1,  2,   and   —  3? 

Ans.  a?3  —  7x  +  6  =  0. 

3.  What  is    the    equation  whose    roots   are    3,    —  4,    2  +  y  3, 
and   2  —  ^/~^l  Ans.  x*  —  3x3  _  15^.2  _|_  49^,  _  12  =  0. 

4.  WTiat  is  the   equation   whose  roots  are    3  +  y  5,    3  —  y  5, 
and    —  6  7  Ans.  x^  —  32x  +  24  =  0. 


300  ELEMENTS    OF    ALGEBRA.  [CHAP,  X. 

5.  What  IS  the  equation   whose  roots   are    1 ,    —  2,   3,    —  4,   5, 
and   —  6  ? 

Ans.  x<^  +  3x^  —  41a;i  —  STcc^  ^  400^2  +  444a;  —  720  =  0. 


6.  What  is  the  equation  whose  roots  are  ....  2  +  y —  1, 
2  -  /-T    and    -  3  ?  Ans.  x^  —  x""-  -  7x  +  15  =  0. 

Of  the  greatest    Common  Divisor. 

285.  The  greatest  common  divisor  of  two  polynomials  is  the 
greatest  polynomial,  with  reference  to  its  exponents  and  co-efh- 
cients,  that   will   exactly  divide  the   proposed  polynomials. 

If  two  polynomials  be  divided  by  their  greatest  common  divisor, 
the  quotients  will  be  prime  ivilh  respect  to  each  other ;  that  is,  they 
will  no  longer  contain  a  common  factor.     Hence, 

Two  polynomials  are  prime  with  respect  to  each  other  when  they 
have  not  a  common  factor.  >» 

Let  A  and  B  be  two  polynomials,  D  their  greatest  common 
divisor,  and    A',  B',  the  quotients   after  division.     Then 

^  =  A^     and     ~  =  B'; 

and  consequently, 

A  =  A'  X  D,     and     B  =  B'  x  D. 

Now,  if  A^  and  5'  have  a  common  factor  d,  then  d  x  D  would 
be  a  common  divisor  of  the  two  polynomials  and  greater  than  D, 
either  with  respect  to  the  exponents  or  the  co-efficients,  which 
would  be  contrary  to  the  supposition. 

Again,  since  D  exactly  divides  A  and  B,  every  factor  of  D  will 
have  a  corresponding   factor  in  both   A   and  B."  Hence, 

1st.  The  greatest  common  divisor  of  two  polynomials  contains  as 
factors,  all  the  prime  factors  common  to  the  two  polynomials, 
and  dons  not  contain  any  others. 

2SG.  We  will  now  show  that  the  greatest  common  divisor  of 
two  polynomials  will  divide  their  remainder  after  one  of  them 
iias  been  divided  by  the  other 

Let  A  and  B  be  two  polynomials,  D  their  greatest  common 
divisor,  and  suppose  A  to  contain  the  highest  exponent  of  the  let- 
ter with  reference  to  which  the  polynomials  A  and  B  are  arranged 


CHAP.  X.]  GREATEST    COMMON    DIVISOR.  301 

Having  divided  A  by  B,  suppose  we  have  a  quotient  Q  and  a 
remainder  R.     We   may  then  write 

A  =  B  X  Q  +  R. 
If  now,  we    divide    both   members    of  the    equation    by   D,  we 

have 

^_  5_  R 

and  since  we  suppose  A  to  be  divisible  by  D,  the  first  member  of 
the  equation  will  be  entire,  and  consequently,  the  second  member 
must  also  be  entire,  since  an  entire  quantity  cannot  be  equal  to 
a  fraction.  But  since  D  also  divides  B,  the  first  term  of  the  sec- 
ond member  is  entire,  and  consequently,  the  second  term  is  also 
entire,  and  therefore,  R  is  exactly  divisible  by  D. 

We  will  now  show  that  if  D  will  exactly  divide  B  and  R,  that 
it  will  also  divide  A.  For,  having  divided  A  by  B,  as  before 
we   have 

A  =  B  X  Q  +  -R,    and  by  dividing  by  D,  we  obtain 
A         B       ^       R 

But  since  we  suppose  B  and  R  to  be  divisible  by  D,  and  know 
Q  to  be  an  entire  quantity,  the  second  member  of  the  equation  is 
entire  :  hence,  the  first  member  is  also  entire,  that  is,  A  is  ex- 
actly divisible  by  D.  ?i  Hence, 

2dly.  The  greatest  common  divisor  of  two  polynomials,  is  the  same 
as  that  which  exists  between  the  least  polynomial  and  their  remainder 
after  division. 

Remark. — If  either  of  the  polynomials  A  or  5  have  a  factor  A^ 
common  to  all  its  terms,  but  not  common  to  the  other  polynomial, 
the  common  divisor  will  be  found  in  that  part  of  the  polynomial 
which  is   multiplied  by  the  factor  A\ 

287.  From  these  principles,  we  have,  for  finding  the  greatest 
common  divisor  of  two  polynomials,  the  following 

RULE. 

I.  Take  the  first  polynomial  and  suppress  all  the  monomial  Jactors 
common  to  each  of  its  terms.  Do  the  same  with  the  second  polyno- 
mial, and  if  the  factors  so  suppressed  have  a  common  divisor,  set  it 
aside  as  forming  a  part  of  the  common  divisor  sought. 


302  ELEMENTS    OF    ALGEBRA.  [CHAP.   X 

II.  Having  done  this,  prepare  the  dividend  in  such  a  manner  that 
its  Jirst  term  shall  be  divisible  bij  the  first  term  of  the  divisor ;  then 
perform  the  division,  and  suppress  in  the  remainder  all  the  factors 
that  are  common  to  the  co-tffcients  of  the  principal  letter.  Then 
tahe  this  remainder  as  a  divisor,  and  the  sccojid  polynomial  as  a 
dividend,  and  continue  the  operation  with  these  polynomials,  in  the 
same  manner  as  with  the  preceding. 

III.  Continue  this  series  of  operations  until  a  remainder  is  ob- 
tiiincd  which  will  exactly  divide  the  preceding  divisor:  this  last  divisor 
will  be  the  greatest  common  divisor ;  but  if  a  remainder  is  obtained 
which  is  independent  of  the  principql  letter,  and  which  will  not  divide 
the  co-efiicients  of  each  of  the  proposed  polynomials,  it  shows  that 
the  proposed  polynomials  are  prime  with  respect  to  each  other,  or 
that  they  have  not  a  common  factor. 

EXAMPLES. 

1.  Find   the  greatest   common   divisor  of  the  polynomials 
a3  _  a-^jj  _|_  3ai2  _  3^3^     and     a^  _  ^^b  +  Ah"^. 

First   Operation.  Second  Operation. 


a3  _  (M  +  3ab^  -  3^3 
4^2j~—    a62_3i3 


|a2  _  5ab  +  4Z»2 
a  +46         ~ 


a2  —  5ab  -f  4F 


—  Aab  +  4*2 


la  —  b 
a-  U 


0. 


1st  rem.    19aZ/2  _  19^3 
or,  19b^{a-b). 

Hence,  a  —  Z>    is  the   greatest  common  divisor. 

We  begin  by  dividing  the  polynomial  of  the  highest  degree  by 
that  of  the  lowest ;  the  quotient  is,  as  we  see  in  the  above  table, 
a  +  4b,  and  the  remainder    Idab"^  —  I9b^. 

But,  19a/y2  —  19Z>3  =  19^2  (a  _  i). 

Now,  the  factor  19^2^  -^yill  divide  this  remainder  without  dividing 
a2  —  5ab  +  4^2  ; 
hence,    the    factor   must   be    suppressed,    and    the   question    is    re- 
duced to  finding  the  greatest  common  di^-isor  between 
a"^  —  5ab  +  462     and     a  —  b. 

Dividing  the  first  of  these  two  polynomials  by  the  second,  there 
is  an  exact  quotient,  a  —  4b ;  hence,  a  —  b  is  the  greatest  com- 
mon divisor  ol  liie  two  given  polynomials.  To  verify  this,  let  each 
be   divided  by  a  —  b. 


ci-i.\i\  X  ■ 


GREATEST    COMMON    DIVISOR. 


303 


2.   Find  the   greatest  common   divisor  of  the  polynomials 

3a^  —  oaW  +  2ai*     and     2a*  —  "iaW  +  M. 
We  first  suppress  a,  which  is  a  factor  of  each  term  of  the  first 
polynomial :    we  then  have 

3a^  —  baW  +  2^-*  II  2a*  —  3aW  +  ¥. 
We   now  find  that  the  first   term  of  the  dividend  will   not   con 
i.iin  the  first  term  of  the  divisor.     We  therefore  multiply  the  divi- 
dend by  2,  which  merely  introduces  into  the  dividend  a  factor  not 
common  to  the  divisor,  and  hence  does  not  affect  the  common  divi- 
sor sought.      We  then  have 

6a*  —  10a2Z»2  -i-  Ah*  jl2a*  —  3a2/;2  +  b* 
6a*  -     9a2<^2  _|_  3^4  I  3 

—       aW  +     b* 

We  find  after  division,  the  remainder  —  a^h"^  +  ¥,  which  we 
put  under  the  form  —  b"^  (a^  —  b"^).  We  then  suppress  —  i^,  and 
divide 


2a*  —  3a2Z/2  -j-  b* 
2a*  —  2a2i2 


Z*2 


2a2  —  ^2 


—  a2^2  _|_  bi 

—  aW  +  i*. 


Hence,    a2  —  b"^   is  the  greatest  common  divisor. 
3.  Let    it    be  required  to    find  the    greatest   common  divisor  be- 
tween the  two  polynomials 

—  3^3  +  3ai2  _  an  +  a^,     and     Ah'^  —  5ab  +  a2. 
rirst   Operation. 


—  V2b^  +  12a52  —  Aa^b  +    4a3 


ist  rem. 

2d  rem. 
or. 


—  3ai2  —    a2^  _j_    4^3 

—  120^2  _  Aa%+  16a3 


1 4P  —  5ab  +  a2 
—  3^,     —  3a 


—  19a2^  +  19a3 
19«2(_  b  +  a). 

Second   Operation. 


4^2  _  ^nh  +  «2 


—     ah  -\-  a2 
'  0^ 


—     i  4-  a 


Ah  +  a 


llcnce,    —  b  ■\-  a,   or    a  —  i,  is  the  greatest  common  divisor. 


304  ELEMENTS    OF    ALGEBRA.  [CHAP.   X 

In  tlie  first  operation  we  meet  with  a  difficulty  in  dividing  tlie 
two  polynomials,  because  the  first  term  of  the  dividend  is  not 
exactly  divisible  by  the  first  term  of  the  divisor.  But  if  we  ob- 
serve that  the  co-efficient  4,  is  not  a  factor  of  all  the  terms  of 
the  polynomial 

4i2  —  5ab  +  a^, 

and  therefore,  by  the  first  principle,  that  4  cannot  form  a  part  w. 
the  greatest  common  divisor,  we  can,  without  affecting  this  com 
mon  divisor,  introduce  this  factor  into  the  dividend.     This  gives 

—  12^*3  +  12ai2  _  4^2^  ^  4^3^ 

and  then  the  division  of  the  terms  is  possible. 

Effecting  this  division,  the  quotient  is  —  3h,  and  the  remainder  is 

—  3ai2  _  a^  +  4«3. 

As  the  exponent  of  b  in  this  remainder  is  still  equal  to  that  of 
b  in  the  divisor,  the  division  may  be  continued,  by  multiplying  this 
remainder  by  4,  in  order  to  render  the  division  of  the  first  term 
possible.     This  done,  the  remainder  becomes 

—  12a52  _  4a^  +  iGa^, 

which  divided  by  4b^  —  5ab  +  «^,  gives  the  quotient  —  3a,  which 
should  be  separated  from  the  first  by  a  comma,  having  no  con- 
nexion with  it.     The   remainder  after  this   division  is 

—  I9a^  +  19a3. 

Placing  this  last  remainder  under  the  form  19a^  (— b -\- a), 
and  suppressing  the  factor  IQa^,  as  lorming  no  part  of  the  com- 
mon divisor,  the  question  is  reduced  to  finding  the  greatest  com- 
mon divisor  between 

4J2  _  5ab  +  a2     and     —  b  -\-  a. 

Dividing  the  first  of  these  polynomials  by  the  second,  we  obtain 
an  exact  quotient,  —  4&  +  a  :  hence,  —  Z>  +  a,  or  a  —  b,  is  the 
greatest  common  divisor  sought. 

288.  In  the  above  example,  as  in  all  those  in  which  the  ex- 
ponent of  the  principal  letter  is  greater  by  unity  in  the  dividend 
than  in  the  divisor,  we  can  abridge  the  operation  by  first  multi- 
plying every   term   of  the    dividend  l)y  the  square   of  the   co-elh- 


CHAP.  X.]  GREATEST    COMMON    DIVISOR.  305 

cient  of  the  first  term  of  the  divisor.  We  can  easily  see  that  by 
this  means,  the  first  partial  quotient  obtained  will  contain  the  first 
power  of  this  co-efficient.  Multiplying  the  divisor  by  the  quotient, 
and  making  the  reductions  with  the  dividend  thus  prepared,  the 
result  will  still  contain  the  co-efficient  as  a  factor,  and  the  division 
can  be  continued  until  a  remainder  is  obtained  of  a  lower  degree 
than  the  divisor,  with  reference  to  the  principal  letter. 
Take  the  same  example  as  before,  viz., 

—  3P  -f  3aP  —  a^b  -\-  a^     and     45^  —  bah  +  d^, 

and  multiply  the  di\'idend  by  the  square  of  4  r=  16  ;  and  we  have 


First   Operation 


120^/2  _      4^2^  _|_    16<j3 


4*2  _  5ah  +  (£'■ 
—  12b  —  3a 


1st.  remainder,  —  19a2*  +  I9a^ 

or,  19a2   (_  J  4-  a). 

Second   Operation. 
4*2  —  oab  -|-  a^  11  —    b  -\-  a 
—    a*  -j-  a2      —  41,  -^  a 
2d  remainder,  —  0. 

289.  When  the  exponent  of  the  principal  letter  in  the  divi- 
dend exceeds  that  of  the  same  letter  in  the  divisor  by  two,  three, 
&c.  units,  multiply  the  dividend  by  the  third,  fourth,  &c.  power 
of  the  co-efficient  of  the  first  term  of  the  divisor.  It  is  easy  to 
see  the  reason  of  this. 

It  may  be  asked  if  the  suppression  of  the  factors,  common  to 
all  the  terms  of  one  of  the  remainders,  is  absolutely  necessary,  or 
whether  the  object  is  merely  to  render  the  operations  more  sim- 
ple. It  will  easily  be  perceived  that  the  suppression  of  these  fac- 
tors is  necessary;  for,  if  the  factor  19a2  was  not  suppressed  in 
the  preceding  example,  it  would  be  necessary  to  multiply  the  whole 
dividend  by  this  factor,  in  order  to  render  its  first  term  divisible 
by  the  first  term  of  the  divisor ;  but  then,  a  factor  would  be  in- 
troduced into  tne  dividend  which  is  also  contained  in  the  divisor  ; 
and  consequently,  the  required  greatest  common  divisor  would  be 
combined  with  the  factor  19a2,  which  forms  no  part  of  it. 

20 


306  ELEMENTS    OF    ALGEBRA.  [CHAP.  X 

290.   For  another  example,  let  it  be  required  to   find  the    great- 
est common  divisor  between  the  two  polynomials, 

a*  +  3a^  +  4a262  —  6aP  +  26*     and     4a^  +  2ai2  _  263, 

or  simply, 

2a2  -j-ab  —  P, 

i^iuce  the  factor  2b  can  be  suppressed,  being  a  factor  of  the  sec- 
(V:iJ   polynomial  and  not  of  the  first. 

First   Operation. 

8«*  +  24a36  +  32^262  _  48^63  +  166* 


+  20^36  +  36rt2^,3  _  48a^3  _|_  ig^i 


2c2  +  a6  —  62 


4a2  +  lOab  +  1352 


+  26^2^2  _  38a^;3  _^  iQiji 


ist  remainder,  —  51a63  +  296* 

or,  —  63  (51a -296). 

Second   Operation. 
Multiply  by  2601,  the  square  of  51. 

5202a2  +  2601a6  —  260162     ,  51^  _    296 
5202«2  _  2958^/6 
1st  remainder,  +  5559a6  —  260162 

5559a6  —  316162 


102a  +  1096 


2d  remainder,  +    56062. 

V     .  . 

The  exponent  of  the  letter  a  in  the  dividend,  exceeding  that  oi 
the  same  letter  in  the  divisor  by  txoo  units,  the  whole  dividend 
is  muUiplied  by  the  cube  of  2  ==  8.  This  done,  we  perform  three 
consecutive  divisions,  and  obtain  for  the  first  principal  remainder, 

—  51a63  +  296*. 

Suppressing  6^,  the  remainder  becomes,  —  51a  +  296  ;  and 
changing  the  signs,  which  is  permitted,  we  have  51a  —  296;  and 
the  new  dividend  is 

2a2  +  a6  —  62. 

Multiplying  the  dividend  b)'-  the  square  of  51  =z  2601,  then  ef- 
fecting the  division,  we  obtain  for  the  second  principal  remainder 
+  56062.  Now,  it  results  from  the  second  principle  (Art.  286), 
iliat  the  greatest  common  divisor  must  be  a  factor  of  the  remain- 
der after  each  division  ;    therefore  it  should  divide  the   remainder 


CHAP.   X.]  GREATEST    COMMON    DIVISOR.  .'i.;? 

560i2.  But  this  remainder  is  indepcndfint  of  the  principal  lellcr  a  : 
hence,  if  the  two  polynomials  have  a  common  divisor,  it  must  be 
independent  of  a,  and  will  consequently  be  found  as  a  factor  in  the 
co-efTicients  of  the  different  powers  of  this  letter,  in  each  of  the 
proposed  polynomials.  But  it  is  evident  that  the  co-efficients  of 
these  polynomials  have  not  a  common  factor.  Hence,  the  two 
given  polynomials  are  prime  with  respect  to  each  other. 

291.  The  rule  for  finding  the  greatest  common  divisor  of  two 
polynomials,  may  readily  be  extended  to  three  or  more  polynomials. 
For,  ha\dng  the  polynomials  A,  B,  C,  D,  &c.,  if  we  find  the  great- 
est common  divisor  of  A  and  B,  and  then  the  greatest  common 
divisor  of  tliis  result  and  C,  the  divisor  so  obtained  will  evidently 
be  the  greatest  common  divisor  of  A,  B,  and  C ;  and  the  same 
process  may  be  applied  to  the   remaining  polynomials. 

Remarks  on   tlic   frrcatest  cunvnon  Divisor. 

o 

292.  Let  A  be  a  rational  and  entire  polynomial,  supposed  to 
be  arranged  with  reference  to  one  of  the  letters  involved  in  it,  a, 
for  example. 

If  this  polynomial  is  not  absolutehj  prime,  that  is,  if  it  can  be 
decomposed  into  rational  and  entire  factors,  it  may  be  regarded 
as  the  product  of  three  principal  factors,  viz., 

1st.  Of  a  monomial  factor  A',  common  to  all  the  terms  of  A. 
This  factor  is  composed  of  the  greatest  common  divisor  of  all  the 
numerical  co-efficients,  multiplied  by  the  product  of  the  literal  fac- 
tors which  are   common  to   all  the   terms  of  the  polynomial. 

2d.  Of  a  polynomial  factor  A",  independent  of  a,  which  is  com- 
mon to  all  the  co-efficients  of  the  different  powers  of  a,  in  the 
arranged  polynomial. 

3d.  Of  a  polj-nomial  factor  A'^\  depending  upon  a,  and  in  which 
the  co-efficients  of  the  different  powers  of  a  are  prime  with  each 
other,  so  that  we  shall  have 

A  =  A'  X  A''  X  A"\ 

Sometimes,  one  or  both  of  the  factors  A\  A",  reduce  to  unity , 
but  the  above  is  the  general  form  of  rational  and  entire  polyno- 
i/iials.     Hence,  their  greatest  common  divisor  may  assume  the  form 


308  ELEMENTS    OF    ALGEBRA.  [CHAP.  X 

D'  denoting  the  greatest  monomial  common  factor,  D'^  the  great- 
est polynomial  factor  independent  of  a,  and  D'^^  the  greatest  poly- 
nomial factor  depending  upon  this  letter. 

In  order  to  obtain  D',  find  the  monomial  factor  A''  common  to  all 
the  terms  of  A.  This  factor  is  in  general  composed  of  literal  fac- 
tors, which  are  fomid  by  inspecting  the  terms,  and  of  a  numerical 
co-efficient,  obtained  by  finding  the  greatest  common  divisor  of  the 
aumerical  co-efficients  in  A. 

In  the  same  way,  find  the  monomial  B''  common  to  all  the  terms 
of  B ;  then  determine  the  greatest  factor    D',    common  to  A'  and  B\ 

This  factor  D'  is  set  aside,  as  forming  the  first  part  of  the  re- 
quired common  divisor.  The  factors  A'  and  B'  are  also  suppressed 
in  the  proposed  polynomials,  and  the  question  is  reduced  to  find- 
ing the  greatest  common  diA^isor  of  two  new  polynomials  which 
do  not   contain  a   common  monomial  factor. 

EXAMPLES. 

1.  It  is  required  to  find  the  greatest  common  divisor  of  the 
two  polynomials 

a2^2  _  (p.j2  _  ^2^2  -f  c^     and     Aa'^d  —  2ac2  +  2c^  —  'iacd. 

The  second  contains  a  monomial  factor  2.  Suppressing  it,  and 
arranging  the  polynomials  with  reference  to   d,  we  have 

(a2  —  c2)  (/2  _  a2c2  _|_  c\     and     {2d^  —  2ac)  d  —  ac'^  +  c^. 

It  is  first  necssary  to  ascertain  whether  there  is  a  common 
divisor  independent  of  d. 

By  considering  the  co-efficients  a^  —  c^  and  —  a^c^  +  c^,  of 
the  first  polynomial,  it  will  be  seen  that  —  a'^c^  +  c*  can  be  put 
under  the  form  —  c^  {cfl  —  c^)  :  hence  a^  —  c^  is  a  common  fac- 
tor of  the  co-efficients  of  the  first  polynomial.  In  like  manner, 
the  co-efficients  of  the  second,  2a'^  —  2ac  and  —  ac"^  +  c^j  can 
be  reduced  to  2a  {a  —  c)  and  —  c"^  (^a  —  c) ;  therefore,  a  —  c  is 
a  common  factor  of  these   co-efficients. 

Comparing  the  two  factors  a^  —  ^2  and  a  —  c,  we  see  that  the 
last  will  divide  the  first ;  hence  it  follows  that  a  —  c  is  a  coin- 
nvin  factor  of  the  proposed  polynomials,  and  it  is  that  part  of 
licir  greatest  common  divisor  which  is  independent  of  d 


CHAP.   X.] 


GREATEST    COMMON    DIVISOR. 


Suppressing  ifi  —  c-  in  the  first  polynomial,  and  a  —  c  in  the 
second,  we  obtain  the  two  polynomials  d"^  —  c^  and  lad  —  c-,  to 
which  the  ordinary  process  must  be  applied 


f/2  _  c2 


\2ad  —  f2 


'lad  +  c2 


+  %icH  —  4a-c2 


—  4«2c2    -j-    c*. 

After  having  multiplied  the  dividend  by  \(fi,  and  performed  two 
consecutive  divisions,  we  obtain  a  remainder  —  'id^c^  +  c*,  inde- 
pendent of  the  letter  d :  hence  the  two  polynomials  d^  —  c^  and 
lad  —  c2,  are  prime  with  each  other.  Therefore,  the  greatest  com- 
mon divisor  of  the  proposed  polynomials  is    a  —  c. 

Again,  taking  the  same  example,  and  arranging  with  reference 
to  a,  it  becomes,  after  suppressing  the  factor  2  in  the  second 
polynomial, 

(,/2  _  ^2)  ,,2  _  c^fp.  ^  c+,     and     2Ja2  _  {^cd  4-  c^)  a  4-  c^. 

It  is  easily  perceived,  that  the  co-efficients  of  the  different  powers 
of  a  in  the  second  polynomial,  are  prime  with  each  other.  In  the 
first  polynomial,  the  co-efficient  —  c^d?  -\-  c*,  of  the  second  term, 
or  of  a",  becomes  —  <?  [^  —  c^) ;  whence,  cZ^  —  c^  is  a  common 
factor  of  the  two  co-efficients,  and  since  it  is  not  a  factor  of  the 
second  polynomial,  it  may  be  suppressed  in  the  first,  as  not  form- 
ing a  part  of  the  common  divisor. 

By  suppressing  this  factor,  and  taking  the  second  polynomial 
for  a  dividend  and  the  first  for  a  divisor  (in  order  to  avoid  prepa- 
ration), we  have 

1st.    2da^  —  2e<i  !  a  +  c3  \\a^  —  c^ 

-    C2 

Rem.    -     —  2cd 


2d 


a  -f-  Idc^ 
+       c3 
or,    -     -      a  —  c, 
by  suppressing  the  common  factor   (—  led 
2d.  a2  -  ( 


c2). 


+  ac 


«  +  c 


0 


After   having    performed  the    first  division,   a    remainder    is    ol>- 
lained    which    contains     —  led  —  c2,    as   a    factor   of    its    two    co- 


310  ELEMENTS    OF    ALGEBRA.  [CHAP.   X. 

efTicients  ;  for  2Jc2  -{-  c^  =z  —  c  (—  2cd  —  c-).  This  factor  being 
suppressed,  the  remainder  is  reduced  to  a  —  c,  which  Avill  exactly 
divide    a^  —  c^. 

Hence,    a  —  c   is  the  required  greatest  common  divisor. 

293.  There  is  a  remarkable  case,  in  which  the  greatest  com- 
mon divisor  may  be  obtained  more  easily  than  by  the  general 
method ;  it  is,  when  one  of  the  two  polynomials  contains  a  letter 
which  is  not  contained  in  the  other. 

In  this  case,  it  is  evident,  that  the  greatest  common  divisor  is 
independent  of  this  letter.  Hence,  by  arranging  the  polynomial 
which  contains  it,  with  reference  to  this  letter,  the  required  com- 
mon divisor  will  be  the  same  as  that  which  exists  between  the  co-ejffi- 
cients  of  the  different  powers  of  the  principal  letter  and  the  second 
polynomial. 

By  this  method  we  are  led,  it  is  true,  to  determine  the  greatest 
common  divisor  between  three  or  more  polynomials.  But  they 
will  be  more  simple  than  the  proposed  polynomials.  It  often  hap- 
pens, that  some  of  the  co-efficients  of  the  arranged  polynomial 
are  monomials,  or,  that  we  can  discover  by  simple  inspection  that 
they  are  prime  with  each  other  ;  and,  in  this  case,  we  are  cer- 
tain that  the  proposed  polynomials   are  prime   with   each   other. 

Thus,  in  the  example  1,  treated  by  the  first  method,  after  having 
suppressed  the  common  factor  a  —  c,  which  gives  the  results, 

(^2  —  c^     and     2ad  —  c^, 

we  know  immediately  that  these  two  polynomials  are  prime  with 
each  other  ;  for,  since  the  letter  a  is  contained  in  the  second  and 
not  in  the  first,  it  follows  from  what  has  just  been  said,  that  the 
common  divisor  must  divide  the  co-efficients  2d  and  —  c^,  which 
is  evidently  impossible ;  hence,  they  are  prime  with  respect  to 
each  other. 

2.  Let  it  be  required  to  find  the  greatest  common  divisor  of  the 
two  polynomials, 

2bcq  +  30/np  +  185c  +  5mpq 
and  4adq  —  A2fg    +  2Aad  —  Ifgq, 

by  the  last  principle. 

We  observe,  in  the  first  place,  that  the  two  polynomials  do  no 
contain  any  common  monomial  factor. 


CHAP.    X..]  GREATEST    COM.MOV     DIVISOR.  311 

Since  q  is  common  to  tlie  two  polynomials,  we  can  arrange 
them  with  reference  to  this  letter,  and  follow  the  ordinary  rale. 
But  h  is  found  in  the  first  polynomial  and  not  in  the  second. 
If  then,  we   arrange  the   first  with  reference   to  b,  which  gives 

{3cq  -\-  1 8c)  b  +  30mp  -f-  5mpq, 

the  required  greatest  common  divisor  will  be  the  same  as  that  whici. 
exists  between  the  second  polynomial  and  the  two  co-efficients 

3cq  +  ISr     and     30/np  -j-  5mpq. 

Now,  the  first  of  these  co-eflicients  can  be  put  under  the  form 
3c  {q  -f-  6),  and  the  other  becomes  5mp  (§'  +  6) ;  hence  ^  +  6  is 
a  common  factor  of  these  co-efiicients.  It  will  therefore  be  suffi- 
cient to  ascertain  whether  7  +  6,  which  is  a  prime  divisor,  is  a 
factor  of  the  second  polynomial. 

Arranging  this  polynomial  Avith  reference   to  q,  it  becomes 

{4ad  --7fg)q-42fg+2iad; 

as  the  second  part,  2iad  —  A2fg  =  6  [Aad  —  Ifg),  it  follows  that 
this  polynomial  is  divisible  by  5'  +  6,  and  gives  the  quotienl 
Add  —  Ifg.  Therefore,  j  -f  6  is  the  greatest  common  divisor  of 
the  proposed  polynomials. 

Remark. — It  may  be  ascertained  that  q-\-6  is  an  exact  divisor 
of  the  polynomial 

{Aad-lfg)q  +  2Aad-A2fg, 

by  a  method  derived  from  the  property  proved  in   Art.   278. 
Make    7  +  6  =  0,   or    7  =  —  6,    in  this  polynomial ;   it  becomes 

{4ad  -7fg)  X  -  6  4-  24ad  -  42fg  =  0  ; 

that  is,  —  6  substituted  for  q  reduces  the  polynomial  to  0 ;  hence 
q  -\-  Q   is  a  divisor  of  this  polynomial. 

This  method  may  be  advantageously  employed  in  nearly  all  the 
applications  of  the  process.  It  consists  in  this,  viz.,  after  obtain- 
ing a  remainder  of  the  first  degree  with  reference  to  a,  when  a 
is  the  principal  letter,  make  this  remainder  equal  to  0,  and  dcdncf 
the  value  of  a  from  this  equation. 

If  this  value,  substituted  in  the  remainder  of  the  2d  divr  ■ 
destroys  it,  then  the  remainder  of  the  1st  degree,  simplified  A  ^ 
292,   is   a   common    divisor.     If  the    remainder  of  the    2d    de>:r;'r 


312  ELEMENTS    OF    ALGEBRA.  [CHAP.   X. 

does  not  reduce  to  0  by  this  substitution,  we  may  conclude  that 
tliere  is  no  common  divisor  depending  upon  the  principal  letter. 

Farther,  having  obtained  a  remainder  of  the  2d  degree,  with 
reference  to  a,  it  is  not  necessary  to  continue  the  operation  any 
farther.     For, 

Decompose  this  "polynomial  into  two  factors  of  the  \st  degree. 
»vhich  is  done  by  placing  it  equal  to  0,  and  resolving  the  result- 
jquation  of  the  2d  degree. 

When  each  of  the  values  of  a  thus  obtained,  substituted  in  the 
remainder  of  the  3d  degree,  destroys  it,  it  is  a  proof  that  the  re- 
mainder of  the  2d  degree,  simplified,  is  a  common  divisor  ;  when 
only  one  of  the  values  destroys  the  remainder  of  the  3d  degree, 
the  common  divisor  is  the  factor  of  the  1st  degree  with  respect 
to  a,  which   corresponds  to  this  value. 

Finally,  when  neither  of  these  values  destroys  the  remainder  of 
the  3d  degree,  we  may  conclude  that  there  is  not  a  common  divi- 
sor depending  upon  the  letter  a. 

It  is  here  supposed  that  the  two  factors  of  the  1st  degree  with 
reference  to  a,  are  rational,  otherwise  it  would  be  more  simple  to 
perform  the  division  of  the  remainder  of  the  3d  degree  by  that  cf 
the  second,  and  when  this  last  division  cannot  be  performed  ex- 
actly, we  may  be  certain  that  there  is  no  rational  common  divisor, 
for  if  there  was  one,  it  could  only  be  of  the  1st  degree  with  re- 
spect to  a,  and  should  be  found  in  the  remainder  of  the  2d  degree, 
which  is  contrary  to  the  hypothesis. 

3.  Find  the  greatest  common  divisor  of  the  two  polynomials 

6x5  —  4,T*  —  lla;3  —  3a;2  _  3a;  _  1 
and  4x4  _|_  2^3  —  IQx^  +  3,r    —  5. 

Ans.  2x'^  —  4x'^  -\-  x  —  1 

4.  Find  the  greatest  common  divisor  of  the  polynomials 

20x6  _  i2oc5  -(_  iG-ci  _  153;3  _|_  ]4j,2  _  15^,  ^  4, 

and  15x4—    9^3  +  47a;2  —  21a;   +  28. 

Ans.  5x"  —  Sa?  +  4. 

5.  Find  the  greatest   common   divisor  of  the   two  polynomials 

5a^^"  +  2a^3  +  ca^    —  3a-b^  +  hca 
aiid  aS  4-  5a^d   —  aW  +  ba%d. 

Ans.  a^  _f_  a5. 


CHAP.  X.]         TRANSFORMATION  OF  EQUATIONS.  313 

Tran.fjormadoii  cf  Equations. 

The  transformation  of  an  equation  consists  in  changing  its  form 
without  affecting  the  equality  of  its  members.  The  object  of  a 
transformation,  is  to  change  an  equation  from  a  given  form,  to 
another  form  that  is  more  easily  resolved. 

First   Transformation. 
To  make  the  Denominators  disappear  from  an  Equation. 
29 i.  If  we  have  an  equation  of  the  form 

x"'  +  Px'"-!  +  Qx'"-2  _(_  ...  r.c  +  Z7=  0, 

and  make  a?  ^  -4- ; 

k 

we  shall  have,  after  substituting  this  value  for  x,  and  multiplying 
every  term  by  k^, 

y""  +  P%'"-i  +  Qk-y^-^  -f  P^3y'"-3+  .  .  .  +  Tk'^-^y  +  f/A"'  —  0, 
an  equation  in  which  the  co-efficieuts  of  y  are  equal  to  those   of  the 
given  equation,  multiplied  respectively  by  yt°,  A\  k"^,  k^,  k*,  &c. 

This  transformation  is  principally  used  to  make  the  denominators 
disappear  from  an  equation,  when  the  co-efficient  of  the  first  term  is 
unity. 

As  an  example,  take  the  equation  of  the  4th  degree, 
a     „         c  e  s 

If  we  make  x  =  -~, 

k 

y  being  a  new  unknown  quantity  and  k  an  indeterminate  quantity, 

we  have 

ak       ^         ck"^     „         ek^  gk* 

Now,  there  may  be  two  cases — 

1st.  Where  the  denominators  b,  d,  f  A,  are  prime  with  each 
other.  In  this  hypothesis,  as  k  is  altogether  arbitrary,  take 
k  =  hdfh,  the  product  of  the  denominators,  the  equation  will  then 
become 

r/  +  adfh  .  y3  +  ch'^dfW  .  y^  +  eb^dfW  .  y  +  gb^d^fP  =  0, 
in   which   the    co-efficients    are  entire,   and    that   of  its    first   term 
unity. 


314  ELEMENTS    OF    ALGEBRA.  [CHAP.   X. 

We  can  determine  the  values  of  x  correspondiog  to  those  of  3, 
from  the  equation, 

_    y 

*  ~  bdfk' 

2d.  When  the  denominators  contain  common  factors,  we  shall 
evidently  render  the  co-efficients  entire,  by  making  k  equal  to  the 
smallest  multiple  of  all  the  denominators.  But  we  can  simplify 
still  more,  by  giving  to  k  such  a  value  that  A\  k-,  P,  .  .  .  shall 
contain  the  prime  factors  which  compose  b,  d,  f,  h,  raised  to  pow- 
ers at  least  equal  to  those  which  are  found  in  the  denom.inators. 

Thus,  the  equation 

5^5^  7  13         „ 

•x* x^  A a;- X =:  0, 

6  12  150  9000 


becomes 


bk    ,       5^2    ^       7A-3  13A* 


after  making    x  =  -^,    and  reducing  the  terms. 

First,  if  we  make  k  =  9000,  which  is  a  multiple  of  all  Ine 
other  denominators,  it  is  clear,  that  the  co-efficients  become  whole 
numbers. 

But  if  we  decompose  6,  12,  150,  and  9000,  into  their  factors, 
we  find 

6  =  2x3,     12  =  22x3,     150  =  2x3x52,     9000  =  23x32x53; 
and  by  simply  making 

i  =  2  X  3  X  5, 
the  product  of  the  different  simple  factors,  we  obtain 

F  =  22  X  32  X  52,     F  =  23  X  33  X  53,     /t*  =  2*  X  34  X  5% 

whence  we  see  that  the  values  of  k,  F,  k^,  k*,  contain  the  prime 
factors  of  2,  3,  5,  raised  to  powers  at  least  equal  to  those  which 
enter  into  6,   12,  150,  and  9000. 
Hence,  the  hypothesis 

A  =  2  X  3  X  5, 

is  sufficient  to  make  the  denominators  disappear.  Substituting  th) 
value,  the  equation  becomes 

^       5.2.3.5     3       5.22.32.52    ^       7.23.33.53  13.2*.3'^.54  _ 

^  2^3^  ^   "*        22;3~~  ^  2.3T52"  ^  23r3'2:53"  ~  ^ 


CHAP.   X.]  TRANSFORMATION    OF    EQUATIONS.  315 

which  reduces  to 

yt  -  5.5y3  +  5.3.5y  _  7.22.32.53/  —  13.2.32.5  =  0  ; 
or  y*  — 25y3  +  375y2  _  I260y  —  1170  =  0. 

Hence,  we  perceive  the  necessity  of  taking  k  as  small  a  number 
as  po.ssible :  otherwise,  we  should  obtain  a  transformed  equation, 
having  its  co-efficients  very  great,  as  may  be  seen  by  reducinc 
the  transformed  (equation  resulting  from  the  supposition   k  =  9000. 

Hence  we  see,  that  any  equation  may  be  transformed  into  another 
equation,  of  which  the  roots  shall  be  a  multiple  or  sub-multiple  of 
those  of  the  given  equation. 

EXAMPLES. 

1.  oc^ :r  3^2  H — -X =  0. 

3  36  72 

y 

Making   x  =  ^-,    and  we  have 
o  6 

y3_  14y2-(-  lly  —  75  =  0. 

5       13    ,       21  32      ^         43  1 

2.  x^  —  -—  a:'  +  -r  a'3 x^ x =  0. 

12       ^40  225  600  800 

Making  x  —  — -^ —  =  -^,    and  we  have 

2-^.3.0         60 

ys  _  55y  _^  I890y3  -  30720y2  -  928800y  +  972000  =  0. 

Second   Transformation. 
To  make  the  second   Term  disappear  from  an  Equation. 
295.  The  difficuUy  of  resolving  an  equation  generally  diminishes 
with  the  number  of  terms  involving  the  unknown  quantity.     Thus 
the  equation 

a?2  =  g,     gives  immediately,     x  =z  zt.  y  q, 
while  the  complete  equation 

x"^  -{- 2px  -^  q  z=i  0, 
requires  preparation  before  it  can  be  resolved. 

Now,  any  given  equation  can  always  be  transformed  into  another 
equation,  in  which  the  second  term  shall  be  wanting. 
For,  let  there   be   the   general   equation 

-jm  ^  p^m    1  _j.   Qy.m-7.  ^    _   _    ^   Tx  +    U  =  0. 

Suppose  X  =  u  +  x', 


316 


ELEMENTS    OF    ALGEBRA. 


[CHAP.  X 


II  being  unknown,  and  x'  an    indeterminate  quantity.     By  substitu- 
ting   u  -\-  a/    for  X,  we  obtain 

{71  +  xJ^+P^u  4-  ^)"'~^  +  Q  ("  +  x'Y'-^ .  .  .  -\-T{u-{-x')-{-U  =  0. 

Developing  by  the  binomial  formula,  and  arranging  according  to 
the  decreasing  powers  of  u.  we  have 


-f  mx 
+  P 


+  m 


m  —  1 


+  (m  -  1)  Pa/ 
+  Q 


+ 


+  a?"» 

+  Pa;""-i 

+  Qx^-^-z 

+     .   .   . 

+  Tx" 

+  f^ 

).  =  0 


Since  a;'  is  entirely  arbitrary,  we  may  dispose  of  it  in  such  a 
way  that  we  shall  have 

P 

mx'  4-  P  =  0  ;     whence,     ,t'  z= . 

m 

Substituting  this  value  of  x'  in  the  last  equation,  we  shall  obtain 
an  equation  of  the  form, 

in  which  the  second  term  is  wanting. 

If  this  equation  were  resolved,  we  could  obtain  any  value  of  x 
corresponding  to  that  of  u,  from  the  equation 

X  ■=  u  -\-  x  ,     or     X  =  u . 

7n 

Whence,  in  order  to  make  the  second  term  of  an  equation  dis- 
appear, 

Substitute  for  the  unknown  quantity  a  new  unknown  quantity,  uni- 
ted with  the  co-ej^cient  of  the  second  term,  taken  with  a  contrary 
sign,  and  divided  by  the  exponent  of  the  degree  of  the  equation. 

Let  us  apply  the   preceding  rule   to  the  equation 

■T^  +  2px  =  q. 

[f  we  make  x  =  u  —  p, 

we  have  (m  —  p)'-*  -{-  2p  [u  —  p)  =  q  ; 


CHAP.  X.]  TRANSFORIIATIOX    OF    EQUATIONS.  317 

and  by  performing  the  multiplications  and  reducing, 

u^  —f  =  q, 
whicti  gives  u  =z  ±  -^  q  -\-  p^  -^ 

and  consequently,  a;=  —  p  ±  -^  q  -{-  p"^. 

296.  Instead  of  making  the  second  term  disappear,  it  may  l)e 
required  to  find  an  equation  which,  shall  be  deprived  of  its  third 
fourth,  or  any  other  term.  This  is  done,  by  making  the  co-efR- 
cient  of  u  corresponding  to  that  term  equal  to  0.  For  example, 
to  make  the  third  term  disappear,  we  make,  in  the  above-transformed 
equation 

TTl  1 

m x"^  +  (»?  -  1)  -P^r^  +  Q  =  0, 

from  which  we  obtain  two  values   for  x^,  which  substituted  in  the 
transformed  equation  reduce     it  to  the  form 

^m  _j_  P'um-\  ^  R^uni-3       _    ^    T'u  +    U'  =   0. 

Beyond  the  third  term  it  will  be  necessary  to  resolve  an  equa- 
tion of  a  degree  superior  to  the  second,  to  obtain  the  value  of  a/  • 
and  to  cause  the  last  term  to  disappear,  it  will  be  necessary  to 
resolve  the  equation 

a/m  +  Par"»-i  .  .  .  +  Tx"  +  U  —  0, 

which  is  what  the  given  equation  becomes  when  a:'  is  substituted 
for  x. 

It  may  happen  that  the  value 

m ' 
which  makes    the  second  term  disappear,  causes    also   the  disap- 
pearance of  the  third  or  some  other  term.     For  example,  in  order 
that  the  third  term  may  disappear  at  the  same  time  with  the  second, 
it  is  necessary  that  the  value  of  x'  which  results  from  the  equation 

m 
shall  also  satisfy  the  equation 

771    —    1 

m x"^  -\-{m-l)Px'  -{-  Q  =  0. 

P 

Now.  if  in  this  last  equation,  we  replace  a/  by ,  Ave  have 

m 

m  —  l    P2  P2 

m- .  — -(/rt-l)  — +  Q  =  0,     or     (m  -  1)  P^ -2mQ  =  0; 

2        Trr  m 


318  ELEMENTS    OF    ALGEBRA. 

and  consequently,  if 


[CHAP.   X. 


P2  = 


2mQ 


m  —  1 

the  disappearance  of  the  second  term  will  also  involve  that  of  the 

the  third. 

Formation  of  derived  Polynomials. 
297.  The  relation 

which  has  been  used  in  the  two  preceding  articles,  indicates 
that  the  roots  of  the  transformed  equation  are  equal  to  those  of 
the  given  equation,  increased  or  diminished  by  a  certain  quan- 
tity. Sometimes  this  quantity  is  introduced  into  the  calculus,  as 
an  indeterminate  quantity,  the  value  of  which  is  afterward  determined 
by  requiring  it  to  satisfy  a  given  condition  ;  sometimes  it  is  a  par- 
ticular number,  of  a  given  value,  which  expresses  a  constant  dif- 
ference between  the  roots  of  a  primitive  equation  and  those  of 
another  equation  which  we  wish  to  form. 

In  short,  the  transformation,  which  consists  in  substituting  u  -\-  x 
for  X,  in  a  given  equation,  is  of  very  frequent  use  in  the  theory 
of  equations.  There  is  a  very  simple  method  of  obtaining,  in  prac 
tice,  the  transformation  which  results   from  this  substitution. 

To  show  this,  let  us   substitute  for  x,    u  +  x'   in  the  equation 

then,  by  developing,  and  arranging  the  terms  according  to  the 
ascending  powers  of  ti,  we  have 


+  P  a;'"*-!  +  (m  —  1 )  Px'^-^ 

+ +  .  .  . 

^Tx"      +T 
+  U 


u-{-  m- 


m  —  1 


1.2 


,771  —  2   „     , 

+  (m  — 1)— -— Pa:""-3 

+  (m-2)^^Qy--* 
+  .  .. 


u^-\-  . .  .u" 


>=0 


If  we  observe  how  the  co-efficients  of  the  different  powers  of  u 
are  composed,  we  shall  see  that  the  co-'sfficient  of  vP,  is  what  the 


Cl'.AP.   X.J  FORMATION    OF    DERIVED    POLYNOMIALS.  319 

tirst  member  of  the  given  equation  becomes  vrhen  x'  is  substitut*>d 
in  place  of  a; ;    we  shall   denote   this  expression  by  X' 

The  co-efficient  of  u^  is  formed  from  the  preceding  term  X', 
by  multiplying  each  term  of  X^  by  the  exponent  of  3/  in  that  term, 
and  then  diminishing  this  exponent  by  unity;  we  shall  denote 
'.bis  co-efficient  by  Y\ 

The  co-efficient  of  ti^  is  formed  from  Y\  by  multiplying  each 
term  of  Y'  by  the  exponent  of  x'  in  that  term,  dividing  the  prod- 
uct by  2,   and  then  diminishing  each  exponent  by  unity.     Repre- 

senting  this   co-efficient  by   -;-,  we  see  that  Z'  is  formed  from  Y^, 

in  the  same  manner  that   Y^   is  formed  from  X' . 

In  general,  the  co-efficient  of  any  power  of  u,  in  the  above- 
transformed  equation,  may  be  found  from  the  preceding  co-efficient 
in  the  following  manner :   viz., 

By  taking  each  term  of  that  co-efficient  in  succession,  multiplying 
it  by  the  exponent  of  x\  dividing  by  the  number  which  marks  the 
place  of  the  co-ejicient,  and  diminishing  the  exponent  of  x'  by  unitij. 

The  law  by  which  the  co-efficients 

-7/  v/ 


1.2'   1.2.3' 

are  derived  from  each  other,  is  evidently  the  same  as  that  which 
governs  the  formation  of  the  terms  of  the  binomial  formula  (Art, 
203).     The  expressions, 

T,    Z\     Y',     TP  .  .  .  . 

are  called  derived  polynomials  of  X\  because  each  is  derived  from 
the  one  which  precedes  it,  by  the  same  law  as  that  by  Avhich  Y^ 
is  deduced  from  X\     Hence,  generally, 

A  derived  polynomial  is  one  which  is  deduced  from  a  given  poly- 
ni  mial,  according  to  a  fxed  and  known  law. 

Recollect  that  X^  is  what  the  given  polynomial  becomes  when 
t     is  substituted  for  x. 

Y'     is  called  the  first-derived  polynomial ; 

Z'     is  called  the  second-derived  polynomial 

V'     is  called  the  third-derived  polynomial 

&c.,  &c 


320  ELEMENTS    OF    ALGEBRA.  [CHAP.  X. 

We  should  also  remember  if  we  mal;e  u  =  0,  we  shall  have, 
a/  =  X,  whence  X'  will  become  the  given  polynomial,  from  which 
the  derived  polynomials  will  then  be  obtained. 

298.  Let  us  now  apply  the  above  principles  in  the  following 

EXAMPLES. 

1.  Let  it  be  required  to  find  the  derived  polynomials  from  ;1. 

equation 

3a;*  +  6x3  —  3a;2  +  2a;  +  1  =  0  =  X 

Now,  u  being  zero,  and  x'  =  x,  we  have  from  the  law  of 
forming  the  derived  polynomials, 

X=  X'  =     3a:*  +    6a;3  —  3a;2  +  2x  +  1  ; 
F  =  12x3+  18x2  — 6x   ^2; 
Z'  =  36x2  +  36:r    _  6  ; 
V  =  72x    +  36  ; 
W  =  72. 
It  should  be  remarked  that  the  exponent  of  x  in  the  terms  1,  2 
—  6,  36,  and  72,  is  equal  to  0  ;    hence,  each  of  those  terms  dis- 
appears in  the  following  derived  polynomial. 

2.  Let  it  be  required  to  cause  the  second  term  to  disappear  in 
the  equation 

x^  —  12x3  +  17x2  _  9a,  _(_  7  ==  0. 

12 

Make  (Art.  295),      x=:m-j =  ?<4-3; 

whence,  x^  =  3. 

The  transformed  equation  will  be  of  the  form 
Z'  V 

^2^2x3 
and  the  operation  is  reduced  to  finding  the  values  of  the  co-efficients 

Z'        V 


X,    T, 


2  '      2.3' 

Now,  it  follows  from  the  preceding  law  for  derived  polynomials, 
that 

X    =       (3)*-12.  (3)3+17.  (3)2-9.  (3)1+7,    or    X^=-110; 
T    =4.(3)3-36.(3)2+34.(3)1-9,    or      -       -     F  =-123: 

^    =6.(3)2-36.(3)1  +  17,    or       -       .       -       -     |^=-37: 

—  =4.(3)1-12 =  0. 

2.3  '^  ^  2.3 


CHAP.   X.]  FORMATION    OF    DERIVED    POLYNOMIALS.  3Vi' 

Therefore  the  transformed  equation  becomes 
u^  —  37«2  _  123«  —  110  =  0. 

3.  Transform  the  equation 

4x^  —  5a:2  +  7x  —  9  =  0 
into   another  equation,  the  roots  of  which  shall    exceed   those    of 
the  given  equation  by  unity. 

Make,  X  =  u  —  1  ;     whence     x^  =  —  1  ; 

and  the  transformed  equation  will  be  of  the  form 

^1.2       '1.2.3 
Hence,  we  have 
K'   =   4.  (-1)3-  5.  (-1)2 +  7.  (-1)1— 9,    or     X'  z=-25j 
y^    =12. (-1)2-10. (-1)1  +  7     -       -       -       -      F   =  +  29; 
Z'  Z' 

^-^^■(-^y-^ Y=-i^^ 

2.3  2.3       ^ 

Therefore,   the  transformed   equation  becomes 
4w3  _  17«2  +  29 w  —  25  =  0. 

4.  What  is  the  transformed  equation,  if  the  second  term  be  made 
«o  disappear  in  the   equation 

a,5  _  lOx*  +  7a;3  +  4a;  —  9  =  0  ? 

Ans.  u^  —  33«3  _  118»2  _  I52u  —  73  =  0. 

5.  What  is  the  transformed  equation,  if  the  second  term  be  made 
4}  disappear  in  the  equation 

3a:3  +  15x2  +  25j:  —  3  =  0  ? 

152 
Ans.  3«3  —  —  =  0. 
9 

6.  Transform  the  equation 

3a;*  —  Ux^  +  7a-2  —  8a;  —  9  =  0 
■jito  another,  the  roots  of  which  shall  be  less  than  the  roots  of  the 

ifiven  equation  by   -;— . 

Ans.  3u-*  —  9u^  —  4«2 u =  0 

9  9 

21 


<i"-^2  ELEMENTS    OF    ALGEBRA.  [CHAP.  X 

P?'oj)er(ics  of  derived  PoJyyiomials. 

299.  ^Ve   will  now   develop   some   of  the   remarkable  properties 
of  derived  polynomials. 

Let  X  =x-^  -{  Px'"-!  +  Qa:'"-2  .  .  .  Tx  -\-U=0 

be   a  given   equation,    and   a,   b,  c,   d,  &c.,  its  m  roots.      We  shall 
then  have   (Art.  281), 

x""  +  Pi'"-'  +  Qx^^-^  .  .  .  ^{x  -a){x~h){x-c)  .  .  .  {x  -  I). 

Making  X  =z  x'  -\-  u, 

nr  omitting  the  accents,  and  substituting  x  -\-  u   for  x,  and  we  have 

(x  +  >/)'"  -f  P  (,r  +  7/)^"-!  +   .  .  .  =  (x  +  w  —  g)  (x  +  w  —  Z-)  .  .  .  ; 

or,  changing  the  order  of  x  and  u,  in  the  second  member,  and  re- 
garding  X  —  (I,  X  —  b,  .  .  .  each  as  a  single  quantity, 


(x-I-M)'"+P(,r  +  (/)"'-i  .  .  .  =:[u+x  —  a)  {u  +  x  —  b)  .  .  .  (m  +  x  — /). 

Now,  by  performing  the  operations  indicated  in  the  two  members, 
we  shall,  by  the   preceding  Article,  obtain   for  the  first  member, 

X  +  Ya  +  —  «^  +  ...  w^ ; 

X  being  the  first  member  of  the  proposed  equation,  and  Y,  Z  .  .  . 
the  derived  polynomials  of  this  member. 

With  respect   to  the  second   member,   it   follows  from  Art.  284, 

1st.  That  the  part  involving  m°,  or  the  last  term,  is  equal  to  the 
product  (x  —  a)[x  —  b)  .  .  .  {x  —  /)  of  the  factors  of  the  proposed 
equation. 

2d.  The  co-efficient  of  u  is  equal  to  the  sum  of  the  products 
of  these  m  factors,  taken  m  —  1   and  m  —  1 . 

3d.  The  co-efficient  of  ifi  is  equal  to  the  sum  of  the  products 
of  these  m  factors,  taken  m  —  2  and  m  —  2  ;    and  so  on. 

Moreover,  since  the  two  members  of  the  last  equation  are  iden- 
tical, the  co-efficients  of  the  same'  powers  are  equal  (240  \    Hence, 

X  =  (x  _  o)  (x  -  h)  (x  _  c)  .  .  .  (x  —  /), 

which  was  already  known.  Hence  also,  Y,  or  the  first-derived 
polynomial,  is  equal  to  the  sum  of  the  products  of  the  m  factors  of 
the  first  degree  in  the  proposed  equation,  taken  m  —  1  and  m  —  1  ; 
or    equal    to  the  sum    of  all   the   quotients    that    can    be    obtained    by 


CHAP.  X.]  EQUAL  ROOTS.  2ZS 

dividing  X  by  each  of  the  m  factors  of  the  frst  degree  in   the  pro- 
posed equation  ;    that  is, 


2i 

Also,    ■ — ,    that  is,   the  second-derived  poHTnonii  il,  divided  by  2, 

is  equal  to  the  sum  of  the  products  of  the  m  factors  of  the  pro- 
posed equation  taken  m  —  2  and  m  —  2,  or  equal  to  the  sum  of 
the  quotients  that  can  be  obtained  by  dividing  X.  by  each  of  the 
factors  of  the  second  degree  ;    that  is, 

Z  X  X  X 

+ 


2        [x  —  a){x~l)       [x  ~  a){x  —  c)  '  '  '  [x  —  k)[x  —  I)' 
and  so  on. 

Of  equal  Roots. 

300.  An  equation  is  said  to  contain  equal  roots,  when  its  first 
member  contains  equal  factors.  When  this  is  the  case,  the  de- 
rived polynomial,  which  is  the  sum  of  the  products  of  the  m  fac- 
tors taken  m  —  I  and  ?/j  —  1,  contains  a  factor  in  its  diflerent 
parts,  which  is  two  or  more  times  a  factor  of  the  proposed  equa- 
tion (Art.  299). 

Hence,  there  must  be  a  common  divisor  between  the  frst  member 
nf  the  proposed  equation,  and  its  frst-derived  polynomial. 

It  remains  to  ascertain  the  relation  between  this  common  divi- 
sor and  the  equal  factors. 

301.  Having  given  an  equation,  it  is  required  to  discover  ivhether 
it  has  equal  roots,  and  to  determine  these  roots  if  possible. 

Let  us  make 

X  =  x'"  +  Px^-^  +  Qx'"-^  -{-...  +  Tx  +  U  z=  0, 

and  suppose  that  the  second  member   contains  n  factors  equal  to 
X  —  a,   n'  factors  equal  to   x  —  b,   n"  factors   equal   to    x  —  c  .  .  .> 
and  also,  the   simple  factors    x  —  p,   x  —  q,    x  —  r  .  .  . ;    we  shall 
then  have, 
A"=  (:r  —  o)"  {x  —  bY{x  —  cy  .  .  .{x  -p)(x  —  q){x  —  r)       (1). 

We  have  seen  that  Y,  or  the  derived  pohmomial  of  X,  is  (he 
!fU7n  of  the  quotients  obtained  by  dividing  X  by  each  of  the  m  fac- 
tvrs  vf  the  frst  degree  in   the  proposed  equation  (Art.  299). 


324  ELEMENTS    OF    ALGEBRA.  [CHAP.  X 

Now,  since  X  contains  n  factors  equal  to  x  —  a,  Ave  shall  have 

n  partial   quotients    equal   to ;    and  the  same   reasoning  ap- 

*  X  —  a 

nlies    to  each  of  the    repeated  factors,    x  —  b,    x  —  c.  .  .  .     M-ore 

over,  we  can  form  but  one  quotient  for  each  simple  factor,  which 

is  of  the  form, 

_X_      _X_         X 

X  —  p'     X  —  g      X  —  r 
Therefore,  the  first-derived  polynomial  is  of  the  form, 

nX         nX       i}"X  X  X  X      ,  _. 

Y= + -  + +  .  .  .  +  — + + +  .  •  .  (2). 

X  —  a      X  —  b      X  —  c  X  —  y;      x  —  q      x  —  r 

Dy  examining  the  form  of  the  value  of  A' in  equation  (1),  it  is 
plain  that 

{x  -  aY-\      {x  -  hY'-\      {x  -  cY"-^  .  .  . 
are  factors  common  to  all  the  terms  of  the  polynomial ;  hence  the 
product 

{x  —  ff)"-!  X  (x  —  i)"'-!  X  {x  —  c)"'^-^  .  .  . 
is   a   common   divisor  of   Y.     Moreover,   it  is   evident  that  it  will 
also  divide  X:  it  is  therefore  a  common  didsor  of  A  and  Y\   and 
it  is   their  greatest  common   divisor. 

For,  the  prime  factors  of  X  are  x  —  a,  x  —  h,  x  —  c  .  .  .,  and 
X  —  p,  X  —  q,  X  —  /•...;  now,  x  —  p,  x  —  q,  x  —  r,  cannot  di- 
vide y,  since  some  one  of  them  will  be  wanting  in  some  of  the 
parts  of  y,  while   it  will   be   a   factor  of  all  the  other  parts 

Hence,   the   greatest  common   divisor  of  A'  and   Y  is 

D  =  (x  -  «)"-'  (.r  -  //)"'-'  {x  -  r)"'^-i  .  .  .  ;    that  is, 

The  grpcilest  coniiiKin  divisor  is  composed  of  the  product  oj  those 
factors  which  enter  two  or  more  times  in  the  given  equation,  each 
raised  to  a  poiver  less  by  unity  than  in  the  primitive  equation. 

302.  From  the  above  we  deduce  the  following  method  for  find- 
ing the  equal  ronis. 

To  discover   whether   an    equation 

A'=  0 
contains   any  equal   roots,  furm    Y  or  the  derived  polyno?niul  of  X ; 
then   seek    fur    the    greatest    common    divisor    hetivccn    X   and    y;    if 
oixi  caimot  be  obtained,  the  equation  has  no  equal  roots,  or  equal 
faciors. 


CHAP.  X.]  EQUAL  ROOTS.  325 

If  we  find  a  common  divisor  D,  and  it  is  of  the  first  degree, 
or  of  the  form  x  —  h,  make  a"  —  h  ■=  0,  whence  x  zrz  h.  We  then 
conclude,  that  the  equation  has  two  roots  equal  to  h,  and  has  but 
one  species  of  rqual  roots,  from  which  it  may  be  freed  by  dividing 
X  by  (.r  -  hy.' 

If  D  is  of  the  second  degree  with  reference  to  x,  resolve  the 
equation  D  ^  0.  There  may  be  two  cases  ;  the  two  roots  will 
be  equal,  or  they  will  be  unequal. 

1st.  When  we  find  D  :=:  (x  —  h)-,  the  equation  has  three  roots 
equal  to  h,  and  has  but  one  species  of  equal  roots,  from  which  it 
can  be  freed  by  dividing  X  by  (x  —  //)•*. 

2d.  When  D  is  of  the  form  (,r  —  //)  (r  —  h'),  the  proposed 
equation  has  tico  roots  equal  to  h,  and  tiro  equal  to  h',  from  which 
it   may  be  freed  by  dividing  A'  by   (x  —  hy  [x  —  h'Y,   or   by  D"^. 

Suppose  now  that  D  is  of  any  degree  whatever  ;  it  is  necessary, 
in  order  to  know  the  species  of  equal  roots,  and  the  number  of 
roots  of  each   species,    to   resolve   cumptetely  the   equation 

D  =  0. 
Then,   every   simple   root   of  D   ivill  be   twice  a   root  of  the  given 
equation ;    every  double   root  of  D  will  be  three   times  a  root  of  the 
given  equation  ;    and  so  on. 
As   to  the   simple   roots  of 

X  =  0, 
we  begin  by  freeing  this   equation  of  the   equal  factors    contained 
in  it,   and  the  resulting  equation,   X'  =  0,    will  make   known   the 
simple  roots. 

EXAMPLES. 

1.  Determine  whether  the   equation 

2x*  —  12x3  +  19^.2  _  6x  +  9  =  0 

contains  equal  roots. 

We   have  for  the   first-derived  polynomial  (Art.  297), 

8x3  _  36^2  _^  28x  —  6. 

Now,  seeking  for  the  greatest  common  dinsor  of  these  poly- 
nomials, we  find 

Z)  =  X  —  3  =  0,     whence     x  =  3  ; 
hence,  the  given  equation  has  two  roots  equal  to  3. 


326  ELEMENTS    OF    ALGEBRA.  [CHAP.  X 

Dividing  its   first  member  by  [x  —  3)^,  we  obtain 

2a;2  +1=0;     whence     a'  =  zh  —  y  —  2. 
The  equation,  therefore,  is  completely  resolved,  and  its  roots  are 
3,     3,     +i-/3^     and      _— /-2. 

2.  For  a  second  example,  take 

a;5  _  2a;*  +  3x^  —  7a;2  -f  8a;  —  3  =  0. 
The  first-derived  polynomial  is 

5a;*  —  8x^  +  9a;2  —  14jc  -f  8 ; 
and  the  common  divdsor, 

3^2  —  2a;  +  1  =  (a;  —  ]  )2 : 

hence,  the  proposed  equation  has  three  roots  equal   to   1. 
Dividing  its  first  member  by 

(x  _  1)3  =  a;3  —  3a;2  -\-3x  —  \, 
the  quotient  is 

-1±V^11 

x^  -{-  X  -\-  3  =  0  ;     whence     x  = — ; 

thus,  the  equation  is  completely  resolved. 

3.  For  a   third  example,  take  the   equation 

a;^  +  5^6  +  6a;5  —  6a;*  —  15a;3  —  3a;2  +  8a;  -f  4  =  0. 
The  first-derived  polynomial  is 

7x6  ^  30^5  -f.  30a;*  —  24x3  _  45^,2  _  6a;  +  8 ; 
and  the  common  divisor  is 

X*  +  3x3  _|_  a;2  _  3a,  _  2. 

The  equation 

X*  +  3x3  +  x2  —  3x  —  2  =  0 

cannot  be  resolved  directly,  but  by  applying  the  method  of  equal 
roots  to  it,  that  is,  by  seeking  for  a  common  divisor  between  its 
first  member   and  its   derived   polynomial 

4x3  +  9x2  _|.  2a;  —  3, 

we  find  a  common  divisor,  x  +  1  ;  which  proves  that  the  square 
©f  X  +  1  is  a  factor  of  x*  +  3x3  ^  ^2  _  ^x  —  2,  and  the  cube 
of  X  -|-  1,   a  factor  of  the  first  member  of  the  given  equation. 


CHAP.   X.]  ELIMINATION.  327 

Dividing 

a:*  4-  3x3  4- a;2  —  3a;  —  2     by     (3,  _f.  1)2  _  j,2  _|.  2a;  +  1, 

we  have  x^  +  x  —  2,  which  being  placed  equal  to  zero,  gives 
the  two  roots  a:  =  1,  x  =  —  2,  or  the  two  factors,  x  —  1  aiul 
X  +  2.     Hence  we  have 

X*  +  3x3  +  x2  —  3x  —  2  =  (x  +  1)2  (.^  _  1)  (a,  _^  2). 
Therefore,  the  first  member  of  the  proposed  equation  is  equal  to 
(x  +  1)3  (x  —  1)2  (x  +  2)2; 
that  is,   the   proposed   equation  has   three  roots   equal  to    —  1,   two 
equal  to   +1,  and  two  equal  to    —  2. 

4.  What  are   the   equal  factors   of  the   equation 

x'  —  7x6  ^  10x5  4-  22x*  —  43x3  —  35x2  ^  43^  +  36  =  0. 
Ans.  (x  —  2)2  (x  —  3)2  (x  +  1  )3  =  0. 

5.  What  are  the  equal  factors  in  the  equation 

x^  —  3x6  ^  9a;5  _  ]  9j;4  _^  27x3  _  33_r2  _|_  27x  —  9  =:  0. 

Ans.   (x  —  1)3  (x2  +  3)2  =  0. 

Elimination. 

303.  To  eliminate  between  two  equations  of  any  degree  what- 
ever, involving  two  unknown  quantities,  is  to  obtain,  by  a  series  of 
operations,  performed  on  these  equations,  a  single  equatiun  which 
contains  but  one  of  the  unknown  quantities,  and  which  gives  all  the 
values  of  this  unknown  quantity  that  will,  taken  in  connexion  with 
the  corresponding  values  of  the  other  unknown  quantity,  satisfy  at 
the  same  time  both  the  given  equations. 

This  new  equation,  which  is  a  function  of  one  of  the  unknown 
quantities,  is  called  the  fnal  equation,  and  the  values  of  the  un- 
known  quantity  found  from  it,  are    called  compatible  values. 

Elimination  by  Means  of  Lideterminate  Multijjlif^rs 

304.  Let  there  be   the   equations 

a  X  -\-  b  y  —  c  =0, 
a'x  -\-  b'y  —  c'  =  0. 

If  we  multiply  the  first  by  m,  and  subtract  the  second  from  tlu' 
product,  we  have 

{ma  —  a')  X  +  (mb  —  b'^  y  —  7«c  4"  c'  =  0  .  .  .  (1 ). 


328  ELEMENTS    OF    ALGEBRA.  [CHAP.  X. 

Now,  since  the  value  of   m  is   entirely  arbitrary,  we    may  give 
it  such  a  value  as  to  render  the  co-efficient  of*  x  zero,  which  gives 


ma  —  a'  z=  0,     whence     m  :=  + 


a 


and  {mb  —  b')7j—  mc  +  c'  =  0 (2). 

Substituting  in   equation  (2)  the  value  of  m,  and  we  have 

a  a'c  —  ac'       acf  —  a'c 

"        a'  a'h  —  ab'       aV  —  a'b 

—  .0  —  0 
a 

Had   we   chosen  to   attribute  to  m  such  a  value   as  to  render  the 

co-efficient  of  y  zero  in   equation  (1),  we   should  have  had 

b' 
mb  —  h'  =  0,     whence     ot  =  -— 

b 

and  [ma  —  a')  x  —  mc  -{-  c'  ■=  0 (3). 

Substituting  in  equation  (3)  the  value  of  m,  we   obtain 

Z       _    ' 
b'"       "        I'c-  be' 


~    y  ,       ab'  —  a'b 

—r-  -a  —  a 
b 

The  above  values  for  x  and  y  are  the  same  as  those  deter- 
mined in  Art.  97.  The  principle  explained  above  is  applicable 
to  three  or  more  equations,  involving  a  like  number  of  unknown 
quantities. 

305.  Of  all  the  known  methods  of  elimination,  however,  the 
method  of  the  common  divisor  is,  in  general,  the  best ;  it  is  this 
method  which  we  are  going  to  develop. 

Let  f(x,  j/)=0  =  A,     and    /  (x,  y)  =  0  =  5, 

be  any  two  equations  whatever,  in  which  /  and  /''  denote  any  func- 
tions of  X  and  y. 

Suppose  the  final  equation  involving  y  obtained,  and  let  us  try 
to  discover  some  property  of  the  roots  of  this  equation,  wliich  may 
serve  to  determine  it. 

L  t  1/  —  a 

he  one  of  the  values  of  y  which  will  satisfy  both  the  given  equations. 
This  is  called  a  compatible  value  of  y.  It  is  plain,  that,  since  this 
value  of  y,  in  connexion  with  a  certain  value  of  x,  will  satisfy  both 
('(piations,  that  if  it  be   substituted  in   thom,  there  will   result  two 


CHAP.   X.]  ELI.MIXATION.  329 

equations  involving  x  alone,  which  will  admil  of  at  least  one  com- 
mon va/iif!  of  X ;  and  to  this  common  value  there  will  correspond 
ft  common  divisor  involving  x  (Art.  279).  This  common  divisor 
will  be  of  the  first,  or  of  a  higher  degree  with  respect  to  x,  ac- 
cording as  the  particular  value  of  y  =  a  corresponds  lo  one  or 
more  values  of  x. 

Reciprocally,  every  value  of  y  which,  substituted  in  the  two  equa- 
tions, gives  a  common  divisor  involving  x,  is  necessarily  a  compati- 
ble value,  because  it  then  evidently  satisfies  the  two  equations  at 
the  same  time  with  the  value  or  values  of  x  found  from  this  com- 
mon  divisor  when  put  equal  to  0. 

306.    We  will  remark,  that,  before  tlie  substitution,  the  first  mem- 
bers   of  the    equittiims    cannot,   in    general,    have    a    common    divisor 
which   is  a   I'uriction  of  one   or  both  of  the    unknown  quantities. 
For,  let  us  suppose  for  a  moment  that   the   equations 
^  =  0,     B  =  0, 
are  of  the  form 

A'  X  Dz^O,     B'  X  D  =:0. 

D  being  a  function  of  x  and  y. 

Making  separately  D  =  0,  we  obtain  a  single  equation  involving 
two  unknown  quantities,  which  can  be  satisfied  with  an  infinite 
number  of  systems  of  values.  Moreover,  every  system  which  ren- 
ders D  equal  to  0,  would  at  the  same  time  cause  A'D,  B'D  to 
vanish,  and  would  consequently  satisfy  the  equations 
A  =  Q     and     B  =  0. 

Thus,  the  hypothesis  of  a  common  divisor  of  the  two  polyno- 
mials A  and  B,  containing  x  and  y,  would  bring  with  it  as  a  con- 
sequence that  the  proposed  equations  were  indeterminate.  There- 
foie,  if  there  exists  a  common  divisor,  involving  x  anJl  y,  of  the 
two  polynomials  A  and  B,  the  proposed  equations  will  be  indeter- 
minate, that  is,  they  may  be  satisfied  by  an  infinite  number  of 
systems  of  values  of  x  and  y.  Then  there  woidd  be  no  data  to 
determine  a  final  eouation  in  y,  since  the  number  of  values  of  y 
is   infinite. 

If  the  two  polynomials  A   and  B  were  of  the  form 
A'  X  D,     B'  X  D, 
D  being   a  function  of  x  only,   we    might  conceive    the   equation 
Z)  -—  0    resolved  with   reference    to    x,   which   would    give    one   or 


330  ELEMENTS    OF    ALGEBRA.  [CHAP.   X 

more   values  for  this  unknown.     Each  of  these  values  subslituteJ 
in  the  equations 

A'  X  D^O  and  B'  x  D  =^  0, 
would  verify  them,  without  regard  to  the  value  of  y,  since  D  must 
e  nothing,  in  consequence  of  the  substitution  of  the  value  of  x 
Therefore,  in  this  case,  the  proposed  equations  would  admit  of 
a  finite  number  of  values  for  x,  but  of  an  infinite  number  of  values 
for  y,  and  then  there  could  not  exist  a  final  equation  in  y. 
Hence,  when  the  equations 

A  =  0,  J5  =  0, 
are  determinate,  that  is,  when  they  admit  only  of  a  limited  nnmhet 
of  systems  of  values  for  a;  and  y,  their  first  members  cannot  have 
for  a  common  divisor  a  function  of  these  unknown  quantities,  un- 
less a  particular  substitution  has  been  made  for  one  of  theso 
quantities. 

307.  From  this  it  is  easy  to  deduce  a  process  for  obtaining  the 
final  equation  involving  y. 

Since  the  characteristic  property  of  every  compatible  value  of 
y  is.  that  being  substituted  in  the  first  members  of  the  two  equa 
tions,  it  gives  them  a  common  divisor  involving  a-,  which  they  had 
not  before,  it  follows,  that  if  to  the  t^vo  proposed  polynomials,  ar- 
ranged with  reference  to  .r,  we  apph''  the  process  for  finding  the 
greatest  common  divisor,  \vq  shall  generally  not  find  one.  But,  by 
continuing  the  operation  properly,  we  shall  arrive  at  a  remainder 
independent  of  x,  but  which  is  a  function  of  y,  and  which,  placed 
equal  to  0,  will  give  the  required  final  equation.  For,  every  value 
of  y  found  from  this  equation,  reduces  to  nothing  the  last  remain- 
der of  the  operation  for  finding  the  common  divisor  ;  it  is,  then, 
such,  thai  substituted  in  the  preceding  remainder,  it  will  render 
this  remainder  a  common  divisor  of  the  first  members  A  and  B. 
Therefore,  each  of  the  roots  of  the  equation  thus  formed,  is  a  com- 
patible value  of  y. 

308.  Admitting  that  the  final  equation  may  be  completely  re- 
solved, which  would  give  all  the  compatible  values,  it  would  after- 
ward be  necessary  to  obtain  the  corresponding  values  of  x.  Now, 
it  is  evident  that  it  would  be  suBinent  for  this,  to  substitute  the 
different  values  of  y  in  the  remainder  preceding  the  last,  put  the 
polynomial  involving  x  which  results  from  it,  eijual  to  0,  and  find 


CHAP.  X.]  ELIMINATION.  331 

from  it  the  values  of  x ;  for  these  polynomials  are  nothing  more 
than  the  divisors  involving  a:,  which  become  common  to  A  and  B. 
But  as  the  final  equation  is  generally  of  a  degree  superior  to 
the  second,  we  cannot  here  explain  the  methods  of  finding  the 
values  of  y.  Indeed,  our  design  was  principally  to  show  that, 
two  equations  of  any  degree  being  given,  we  can,  without  supposing 
the  resolution  of  any  equation,  arrive  at  another  equation,  containi?}g 
only  one  of  the  unknown  quantities  which  enter  into  the  proposed 
equations. 

EXAMPLES. 

1.  Having  given  the  equations 

A  —  x^  -{■  xy  +  y"^  —  \  =  0, 
B  =  x^  +  y3=0, 

to  find  the  final  equation  in  y. 


First   Operation. 


x^  +  y^ 

x^  -+■  yx"^  -\-  [y"^  —  ])  X 


x"^  -{-  xy  -\-  y 
X  —  y  =  Q 


—  yx^  —  (y2  —  l^  X  -{-  y^ 

—  yx'^  —  y'^x  —  y-^      -{-  y 

R  =  X  -\-  2y3  —  y  =  1st  remainder. 

Second   Operation. 
a;2  +    yx    +  y2  _  1  JU4.    2y3  - 


x2  +  (2y3  —  y)x  a:  -  (2y3  —  2y) 

—  (2y3  —  2y)  X  +     y^  _  1 

—  (2y3  —  2y)  X  —  4y^  +  6y*  —  2y2 

R^  =  4y'5  —  6y^  +  3y2  —  1  =  2d  remainder 
Hence,   the   final   equation   in  y,   is 

4y6  _  Qyi  4-  3y2  —  1  =  0. 

If  it  were  required  to  find  the  final  equation  in  x,  we  observe 
that  X  and  y  enter  in  the  same  manner  into  the  original  equations ; 
hence,  x  may  be  changed  into  y  and  y  into  x,  without  destroying 
the  equality  of  the  members.     Therefore, 

4x6  _  Q^i  +  3x2  —  1  =  0 
is  the  final  equation  in  x. 


332 


ELEJIENTS    OF    ALGEBRA. 


[CHAP.  X. 


2.  Find  the   final   equation  in  y,  from  the   equations 

A  =  x^  —  Si/x"^  +  (3y2  —  y  +  1 )  ^  —  y^  -\-  rp-  —  2y  =  0, 
B  zzz  x^  —  2i/x   +      y2  —  y  ;_  0. 


First   Operation. 
^3  _  3y^2  _|.  (3^2  _  y  +  1)  ^  _  y3  ^  y2  _  2y  II  a:2  —  2jy  +  /  _  y 
X"  —  2yx'^  -\-     (y2  —  y)  j; 

—  yx2  +    (2y2   +    1)  a;  _  y3  _|.  y2   _   2y 

—  ya.'2  +    2y^a?  —  y^  -{■  y^ 


v  =  Q 


X  —  2y  =  R. 

Second   Operation. 

x"^  —  2xy  +  y"  —  y 
x^  —  23:y 


2y 


Hence, 


r 


x^Q' 

f  —  y  =Q 

is  the  final  equation  in  y.     This  equation  gives 

y  =  1      and     y  =  0. 

Placing  the  preceding  remainder   equal  to  zero  (Art.  308),  and 
substituting  therein  the  values  of 

y  =  1  and  y  =  0, 
we   find   for  the   corresponding  values  of  x, 

X  —  2  and  x  =  0; 
from  which  the  given  equations  may  be  entirely  reserved. 


CHAF.   XI. 1  RESCiLUTIOX    OF    NUMERICAL    EtiUATIONS.  333 


CHAPTER  XL 

RESOLUTION    OF    NUMERICAL    EQUATIONS     INVOLVING    ONE     OR    MORE 
UNKNOWN    QUANTITIES. 

309.  The  principles  established  in  the  preceding  chapter,  are 
applicable  to  all  equations,  whether  their  co-efficients  are  numeri- 
cal or  algebraic,  and  these  principles  are  the  elements  which  are 
to  be  employed  in  the  resolution  of  equations  of  the  higher  de- 
grees. 

It  has  been  already  remarked,  that  analysts  have  hitherto  been 
able  to  resolve  the  general  equations  only  of  the  third  and  fourth 
degrees.  The  general  formulas  which  haA'e  been  obtained  for  the 
resolution  of  algebraic  equations  of  the  higher  degrees,  are  so 
complicated  and  inconvenient,  even  when  they  can  be  applied, 
that  the  problem  of  the  resolution  of  algebraic  equations,  of  any 
degree  Avhatever,  may  be  regarded  as  more  curious  than  useful. 

Therefore,  analysts  have  principally  directed  their  researches  to 
the  resolution  of  numerical  equations,  that  is,  to  those  which  arise 
from  the  algebraic  translation  of  a  problem  in  which  the  given 
quantities  are  particular  numbers.  Methods  have  been  found,  by 
means  of  which,  the  roots  of  a  numerical  equation  of  any  given  de- 
gree, may  always  be  determined. 

It  is  proposed  to  develop  these  methods  in  this  chapter. 

To  render  the  reasoning  general,  we  will  represent  the  proposed 
equation  by 

X  =1  x""  +  Px""-!  +  Qj;'"-2  +  .  .  .    =0. 

in  which  P,  Q  .  .  .  denote  particular  numbers  which  are  real,  and 
either  positive  or  negative. 


334  ELEMENTS    OF    ALGEBRA.  [CHAP.  XT 

First  Principle. 

310.  If  we  substitute  for  x  a  number  a,  and  denote  by  A  what 
X  becomes  under  this  supposition ;  and  again  substitute  a  -\-  u  for 
X,  and  denote  the  new  polynomial  by  A' :  then,  u  may  he  taken 
so  small,  thai  the  difference  between  A''  and  A  shall  he  less  than 
any  assignable  quantity. 

If  now,  we  denote  hy  B,   C,  D, what  the   co-efEcients 

Z         V 
Y,    —  ,     (Art.  297),   become,    when    we    make    a?  =  a,    we 

shall  have  for  the  polynomial  X,  under  the  supposition  that  a  +  « 
is  substituted  for  x, 

equal  to  A  -\- u{B -\-  Cu   +  Du"^  +  .  .  .  u'"-^)  =  A\ 

Now,  the  quantity 

M  (5  +  Cw  4-  Du^  4-  .  .  ■  w"'~0 

is  the  difference  between  A''  and  A  J  s-'^i  it  is  required  to  show 
that  this  difference  may  be  rendered  less  than  any  assignable  quan- 
tity, by  attributing  a  value  sufficiently  small  to  Ji. 

Let  us  take  the  most  unfavorable  case  that  can  occur,  viz.,  let 
us  suppose  that  every  co-efficient  is  positive,  and  that  each  is  equal 
to  the  largest,  which  we  will  designate  by  K.     Then, 

Zm  (1  +  ?i  4-  w^  +  •  •  •  W"'^)  =  u{B  -hCu+  .  .  .  +  M^-i) ; 
and  in  any  other  case, 

Ku  (1  4-  w  +  m2  +  .  .  .  u'»-i)  >  w  (^  4-  C«  -f  .  .  .  w-"-'). 

But  we  have,  by  Art.  61, 

(1  u'"\ 
)  ; 
1  —  u  / 

Ku     ^^  .  ^     Ku 

and (1  —  u^)  < 


1  —  u  1  —  u 

when  «  <  1. 

This  being  premised,  if  we  wish  the  difference  between  A'  and 

A  to  be  less  than  any  number  N,  let  us   make   u  such,  that 

Kn  ^  AT         I.-  1  -1  ^        ^ 

. —    or    <^  iV      which  requires  that,    u  z=  or    <     _ ,  , 

1  —  w  A  +  A 


CHAP.   XI.]  RESOLUTION    OF    NUMERICAL    EQUATIONS.  335 

and  anv  value  of  u  which  will  fulfil  this  last  condition,  will  satisfy 
the  inequality 

iTzi  (1  +  u  +  u2  +  .  .  .  u'"-i)  <  N, 
and  consequently,  render 

«  (5  -(-  Cu  +  Brfl  +  .  .  .  ti'"-^)  <  iV; 

in  which  the    inequality  is    greater   even    than   in   the    expression 
above. 

Second  Principle. 

311.  If  two  numbers  p  and  q,  substituted  in  succession  in  the 
place  of  X  in  a  numerical  equation,  give  two  results  affected  with 
contrary  signs,  the  proposed  equation  contains  a  real  root,  compre- 
hended between  these  two  numbers. 

Let  us  suppose   that  p,  when  substituted  for  x  in   the  equation 
X  =zO,     gives     +  R, 
and   that  q  substituted  in  the   equation 

X=0,     gives     —  R'. 

Let  us  now  suppose  x  to  vary  between  the  values  of  p  and  q 
by  so  small  a  quantity,  that  the  difference  between  any  two  cor- 
responding consecutive  values  of  X  shall  be  less  than  any  assign- 
able quantity ;  in  which  case,  we  say  that  X  is  subject  to  the 
law  of  co7itinuity,  or  that  it  passes  through  all  the  intermediate 
values  between  R  and   —  R\ 

Now,  a  quantity  which  is  constantly  finite,  and  subject  to  the 
law  of  continuity,  cannot  change  its  sign  from  positive  to  nega- 
tive, or  from  negative  to  positive,  without  passing  through  zero : 
hence,  there  is  at  least  one  number  between  p  and  q  which  will 
Siitisfy  the  equation 

X  =  0, 

Knd    consequently,   one    root  of  the   equation    lies    between    these 
numbers. 

312.  We  have  shown  in  the  last  article,  that  if  two  numbers  be 
substituted,  in  succession,  for  the  unknown  quantity  in  any  equation, 
and  give  results  affected  with  contrary  signs,  that  there  will  be 
at  least  one  real  root  comprehended  between  them.  We  are  not, 
'^rv/e  ?r,   to   conclude  that  there   may  not  be  more   than  one  ;  nor 


336  ELEMENTS    OF    ALGEBRA.  [CHAP.   XI. 

that  the  substitution,  in  succession,  of  two  numbers  which  include 
roots  of  the  equation,  will  necessarily  give  results  afl'ected  with 
contrary  signs. 

Third  Prhiciple. 

313.  When  an  uneven  number  of  the  real  roots  of  an  equation 
is  comprehended  between  two  numbers,  the  results  obtained  l<y  sub- 
stituting these  numbers  in  succession  for  x,  will  be  affected  with  cun- 
trary  signs;  hut  if  they  comprehend  an  even  number  of  roots,  the 
results  obtained  by  their  substitution  will  be  affected  with  the  same 
sign. 

To  make  this  proposition  as  clear  as  possible,  denote  by  a,  b, 
c,  .  .  .  those  roots  of  the  proposed   equation, 

:\:  =  o, 

which  are  supposed  to  be  comprehended  between  p  and  q,  and 
by  Y,  the  product  of  the  factors  of  the  first  degree,  with  reference 
to  X,  corresponding  to  the  remaining  real  roots  and  to  the  imagin- 
ary roots  of  the  given  equation. 

The  first  member,  X,   can   then  be   put  under  the  form 

(x  —  a)  {x  —  b)  {x  —  c)  .  .  .   X   Y  =z  X. 

Now,  substituting  p  and  q  in  place  of  x,  we  shall  obtain  the 
two  results 

^p^a){p-b){p-c)...   X   Y\ 

{q-a){q-b){q-c)...X  Y'\ 

Y'  and  Y"  representing  what  Y  becomes,  when  we  replace  in 
succession,  a;  by  />  and  q.  These  two  quantities  Y'  and  Y" ,  are 
afTected  with  the  same  sign ;  for,  if  they  were  not,  by  the  second 
principle  there  would  be  at  least  one  real  root  comprised  between 
p  and  q,  which  is  contrary  to  the  hypothesis. 

To  determine  the  signs  of  the  above  results  more  easily,  divide 
the  first  by  the  second,  and  we  obtain 

(p-«)(p-Z>)(p-c)...  X  F 
(y_a)(y-Z.)(y-c)...   X   r-' 

which  can  be  written  thus. 


p  —  a         p  —  b  p  —  c 

X r-  X  ~ 

q  —  a  q  —  b  q  —  c 


y/, 


CHAP.   XI.]  LIMITS    OF    REAL    ROOTS.  ii'l7 

Now,  since  the  roots  a,  b,  c,  .  .  .  are  comprised  between  p  av.d 
q,  we  have 

p         a,    h,    c,    d  .  .  ., 

and  9         a,    h,    c,    d  .  .  .; 

whence  we  deduce 

p  —  a,  p  —  b,   p  —  c,  .  .  0, 

ajnd  q  —  a,    q  —  b,   q  —  c,  .  .  .       0. 

Hence,  since  p  —  a  and  q  —  a  are  affected  with  contrary  signs, 
as  well  as  p  —  b  and  q  —  b,  p  —  c  and  q  —  c  .  .  .,  the  partial 
quotients 

p  —  a        p  —  b        p  —  c 


q  —  a         q  —  o 


-,  Sic, 


Y' 

are  all  negative.     Moreover,    -^7-^   is  essentially  positive,  since   Y 

and   Y"  are   affected  with  the   same  sign  ;    therefore,   the  product 

p  —  a         p  —  b         p  —  c  Y' 

X  IT-  X  X  .  .  .     -jT-,,  , 

q  —  a  q  —  0  q  —  c  1" 

will  be  negative,  when  the  number  of  roots,  a,  b,  c  .  .  ..  compre- 
hended between  p  and  q,  is  uneven,  and  positive  when  the  number 
is  even. 

Consequently,  the  two  results 

{p-a){p-b){p-c)  .  .  .  X  F, 

and  (?  —  o)  (?—*)(?  —  '^)  ••  •  X  F'^ 

will  have  contrary  or  the  same  signs,  according  as  the  numbei 
of  roots  comprised  between  p  and  q  is  uneven  or  even. 

Limits  of  Real  Roots. 

314.  The  different  methods  for  resolving  numerical  equations, 
consist,  generally,  in  substituting  particular  numbers  in  the  pio- 
posed  equation,  in  order  to  discover  if  these  numbers  verify  it,  or 
whether  there  are  roots  comprised  between  them.     But  by  reflect- 

22 


338  ELEMENTS    OF    ALGEBRA.  [CHAP.   XI 

ing  a  little  on  the  composition  of  the  first  member  of  the  general 
equation 

X'n  -f  Pa;"'-!  +  Qa;'"-2  ...-}-  Tx  -\-  U  =  0, 

we  become  sensible,  that  there  are  certain  numbers,  above  ^^hicb 
it  would  be  useless  to  substitute,  because  all  numbers  above  a 
certain   limit,  would  give  positive  results. 

315.  Let  it  now  be  required  to  resolve  the  following  question: 
To  determine  a  number,  ivhich  suhstitaled  in  place  of  x  will  ren- 
der the  first   term   x"*  greater   than    the    arithmetical  sum   of  all   the 
other  terms. 

Suppose  all  the  terms  of  the  equation  to  be  negative,  except 
the  first,  so  that 

^m  _  p^m-l   _   Qx'n-2   _   _   ^   fx  —    U  =  0. 

It  is  required  to  find   a  number  for  x  which  will  render 

a;'"  >  P:>;"'-i  +  Qa;"'-^  +...-{-  Tx  +  U. 
Let  k  denote  the  greatest  co-efficient,  and  substitute  it  in  place 
of  the  co-efficients  ;    the  inequality  will  then  become 

y,m  y  Ji^m-l   _^   ^^m-2   ^_    .    .    _    _j_   ^jj  ^   ^_ 

It  is  evident  that  every  number  substituted  for  x  which  will 
satisfy  this  condition,  will  for  a  stronger  reason,  satisfy  the  pre- 
ceding.    Now,  dividing  this  inequality  by  x"*,  it  becomes 

k  k  k  k  k 

k 
Making    x  =  k,    the   second   member   becomes    — -  =  1    plus    a 

k 

series  of  positive  fractions.  The  number  k  will  therefore  not  sat- 
isfy the  inequality;  but  by  supposing  a;  = /t  +  1,  we  obtain  for 
the  second  member  the  series  of  fractions 

K  fC  fi  K  rC 

Y^  '^  (M^Tp  "^  {k  +  1)3  +  •  ■  •  +  (^  +  I  )"•-!  +  {k+  i)-"' 

which,  considered  in  an  inverse  order,  is  ar.  increasing  geometri- 

k 

cal    progression,    the    first    term   of  which    is — ,    the   ratio 

^     ^  '  (A  +  1 )'» 

k 

k  -{-  1,  and  the  last  term    — ;    hence,  the   expression   for   tl  e 

k  -f-  i 

sum  of  all  the  terms  is  (Art.  192), 


CHAP.   XI.]  ORDINARV    LIMIT    OF    POSITIVE    ROOTS.  339 

A__  k 

r+T  ■  ^   +  ^^ "~  (TTTp  1 


which  is  evidently  less  than  unity. 

Now,  any  number  >  (^  +  1),  put  in  place  of  x,  will  render  the 

k         k 

sum  of  the  fractions 1 +  •  •  •  still  less.     Therefore, 

a;         x^ 

The  greatest  co-efficient  plus  unity,  or  any  greater  number,  being 
substituted  for  x,  icill  render  the  jirst  term  x"*  greater  than  the  arith- 
metical sum   of  all   the  other   terms. 

316.  Every  number  which  exceeds  the  greatest  of  the  posi- 
tive roots  of  an  equation,  is  called  a  superior  limit  of  the  positive 
roots. 

From  this  definition,  it  follows,  that  this  limit  is  susceptible  of 
an  infinite  number  of  values.  For,  w^hen  a  number  is  found  to  ex- 
ceed the  greatest  positive  root,  every  number  greater  than  this, 
is  also  a  superior  limit. 

But  since  the  largest  of  the  positive  roots  will,  when  substituted 
for  X,  merely  reduce  the  first  member  to  zero,  it  follows,  that 
we  shall  be  sure  of  obtaining  a  superior  limit  of  the  positive  roots 
by  finding  a  number,  which,  substituted  in  place  of  x  renders  the  frst 
member  positive,  and  ivhich  at  the  same  time  is  such,  that  every 
greater  number  will  also  give  a  positive  result. 

Hence,  the  greatest  co-rfficient  of  x  plus  unity,  is  a  superior  limit 
of  the  positive  roots. 

Ordinary  Limit  of  the  Positive  Roots. 

317.  The  limit  of  the  positive  roots  obtained  in  the  last  article, 
is  commonly  much  too  great,  because,  in  general,  the  equation 
contains  several  positive  terms.  We  will,  therefore,  seek  for  a 
limit  suitable  for  all  equations. 

Let  a"*"""  denote  that  power  of  x,  corresponding  to  the  first  neg- 
ative term  which  follows  x^,  and  let  us  consider  the  most  unfavor- 
able case,  viz.,  that  in  which  all  the  succeeding  terms  are  negative 
and  affected  with  the  greatest  of  the  negative  co-efficients  in  the 
equation. 

Let   S  denote   tliis   co-efiicient.     What  conditions  will  render 
j,m  y  Sa-"'-"  +  5'a;'"-"-'  +  .  .  .  Sx  +  S^. 


340  ELEMENTS    OF    ALGEBRA.  [CHAP.   XI. 

Dividing  both  members  of  this  inequality  by  a?'",  we  have 
5  -S  -S  ,      S  S 

Now,  by  supposing 

X  =  \/S  -\-  1,    or  for  simplicity,  making    "y/  S  =  S' 
which  gives,  >S  =  *S''",     and     a?  =  <S'  +  1, 

;he  second  member  of  the  inequality  will  become, 

{S'-\-  1)»  "^  (S'  +  l)»+i  +  •  •  •  +  {S'  +  l)"*-!  "^  (S^  +  !)"•' 
which   is   a  progression  by  quotients,  of  which  is   the 

first  term,  S'  +  1    the  ratio,  and  the  last  term.     Hence, 

the  expression  for  the  sum   of  all  the   terms  is   (Art.   192), 


;S'  +  1  -  1  (S'  +  1)"-!       {S'  +  l)-"' 

which  is  evidently  less  than   1. 

Moreover,    every   number    >  .S'  4-  1    or    yS  +  1,   will,  when 
substituted  for  x,  render  the  sum   of  the  fractions 
S  S 

X"       a'''+i  "T-  •  •  •  • 

still   smaller,   since   the   numerators    remaining   the   same,   the    de 
nominators  will  increase. 


Hence,  y  *S  +  1,  and  any  greater  number,  will  render  the  first 
term  a;'"  greater  than  the  arithmetical  sum  of  all  the  negative  terms 
of  the  equation,  and  will  consequently  give  a  positive  result  for 
the  first  member.     Therefore, 

Unity  increased  by  that  root  of  the  greatest  negative  co-efficient 
whose  index  is  the  number  of  terms  which  precede  the  first  negative 
term,  is  a  superior  limit  of  the  positive  roots  of  the  equation.  If  the 
co-efficient  of  a  term  is  0,  the  term  must  still  he  counted. 

Make  «  =  1,  in  which  case  the  first  negative  term  is  the  sec- 
ond term  of  the  equation  ;  the  limit  becomes 

V^  +  1  =  S  +  1 ; 

I'iiat.  is,  the  greatest  negative  co-efficient  plus  unity. 


CHAP.   XI.]  SMALLEST    LIMIT    IN    ENTIRE    NUMBERS.  341 


Let  ra  r=  2;  then,  the  limit  is  y  <S  +  1.  When  n  =  3,  the 
limit  is    y  /S  +  1. 

EXAMPLES. 

i.  What  is  the  superior  limit  of  the  positive  roots  in  the  equation 
X*  —  5x^  +  37a;2  _  3x  +  39  =  0. 

Ans.  V^+  1  =  V~^+  1  =  Q* 
2    What  is  the  superior  limit  of  the  positive  roots  in  the  equation 
a;5  +  7x*  —  ]2a:3  —  49x^  +  52a;  —  13  =  0. 

Ans.  \/~S  +  1  =  ^49  +1=8. 
3.   What  is  the  superior  limit  of  the  positive  roots  in  the  equation 

X*  -\-  lla;2  —  2507  —  67  =  0. 
In  this  example,  we  see  that  the  second  term  is  wanting,  that  is, 
its  co-efficient  is  zero ;  but  the  term  must  still  be  counted  in  fix- 
ing the  value  of  n.  We  also  see,  that  the  largest  negative  co-efS- 
cient  of  x  is  found  in  the  last  term  where  the  exponent  of  x  is 
zero.     Hence, 

V^  +  1  =  V67  +  1  ; 
and  therefore,  6  is  the  least  whole  number  that  will  certainly  ful 
fil  the  conditions. 

Smallest  Limit  in  Entire  Nutnbers. 

318.  In  Art.  316,  it  was  shown  that  the  greatest  co-efficient  of 
X  plus  unity,  is  a  superior  limit  of  the  positive  roots.  In  the  last 
article  we  found  a  limit  still  less  ;  and  we  now  propose  to  find 
the   smallest  limit  in  whole  numbers. 

Let  X  —  0, 

be  the  proposed  equation.     If  in  this  equation  we  make  x=zx^-\-  u, 
x'  being  indeterminate,  we  shall  obtain  (Art.  297), 
Z' 

X'  +    Y'U   +   — m2  -I-    .   .   .    _|_  j^m  ^  0         (1). 

Let  us  suppose,  that  after  successive  trials  we  have  determine  1 
a  number  for  x' ,  which  substituted  in 

X',    T,    f  .  .  .. 


342  ELEMENTS    OF    ALGEBRA.  [CHAP,  XI 

renders  all  these  co-efficients  positive  at  the  same  time  ,  this  num- 
ber will  be  greater  than  the  greatest  positive  root  of  the  equation 

For,  if  the  co-efficients  of  equation  (1)  are  all  positive,  no  posi- 
ive  number  can  verify  it ;  therefore,  all  the  real  values  of  u 
must  be  negative.     But  from  the   equation 

,  X  =  x^  -{-  u,     we  have     m  =  a;  —  x^ ; 

and  in  order  that  every  value  of  u,  corresponding  to  each  of  the 
values  of  x  and  x',  may  be  negative,  it  is  necessary  that  the  great- 
est positive  value  of  x  should  be  less  than  the  value  of  x'. 

EXAMPLES 

Let  X*  —  5x^  —  6x2  _  I9x  -f  7  =  0. 

As  x^  is  indeterminate,  we  may,  to  avoid  the  inconvenience  of 
writing  the  primes,  retain  the  letter  x  in  the  formation  of  the  de 
rived  polynomials  ;    and  we  have 

X     =z    X*  —    5x^  ~    6a;2  —  .19x  4- 7, 
Y     =  4a:3  —  15x2  — 12a;    -  19, 

—     =r  6a;2  —  ISa;    —  6, 

V 
— — —  z=  4x    —  5. 
2.3 

The  question  is  now  reduced  to  finding  the  smallest  entire  num- 
ber which,  substituted  in  place  of  x,  will  render  all  of  these  poly- 
nomials positive. 

It  is  plain  that  2  and  every  number  >  2,  will  render  the  poly- 
nomial of  the  first  degree  positive. 

But  2,  substituted  in  th(5  polynomial  of  the  second  degree,  gives 
a  negative  result ;  and  3,  or  any  number  >  3,  gives  a  positive 
result. 

Now  3  and  4,  substituted  in  succession  in  the  polynomial  of 
the  third  degree,  give  negative  results ;  but  5,  and  any  greater 
number,  gives  a  positive  result. 

Lastly,  5  substituted  in  X,  gives  a  negative  result,  and  so  does 
6 ;  for  the  first  three  terms,  x*  —  5x^  —  6x^,  are  equivalent  to  the 
expression  x^(x  —  5)  —  6^2,  which  reduces  to  0  when  x  —  6  ;  but 
X  =  7    evidently    gives    a    positive    result.      Hence   7,  which  here 


CHAP.   XI.]  SUPERIOR    LI.MIT    OF    NEGATIVE    ROOTS.  343 

Stands  for  x\  is  a  superior  liinit  of  the  positive  roots  of  the  giinn 
equation.  Since  it  has  been  shown  that  6  gives  a  negative  re- 
sult, it  follows  that  there  is  at  least  one  real  root  between  6  and  7. 

2.  Applying  this  method  to  the  equation 

x^  —  3x*  —  8x3  _  25x^  +  4a:  —  39  =  0, 
i,he  superior  limit  is  found  to  be  G. 

3.  We  find  7  to  be  the  superior  limit  of  the  positive  roots  of 
the  equation 

x5  _  5x*  —  13x3  _(_  i7a;2  _  69  =  0. 

This  method  is  seldom  used,  except  in  finding  incommensurable 
roots. 

Superior  Limit    of  negative    Roots. — Inferior   Limit    of  posi- 
tive and  negative  Roots. 

319.  Having  found    the   superior  limit  of  the    positive   roots,   it 
only  remains   to  find  the  inferior  limit,  and  the  superior  and  infe- 
rior limits  of  the  negative  roots. 
Let,     L     =  superior  limit  of  positive  roots. 
L'    =  inferior  limit  of  positive  roots. 

U^  =  superior  limit  (that  is,  numerically)  of  negative  roots 
t  1/'''  =:  inferior  limit  of  negative  roots. 

1st.  If  in  anv  equation   X  =  0,  we  make    x  =  — ,    we   have   a 

V 

derived  equation   1"=  0.     We  know  from  the  relation  x  =  — ,  thai 

y 

the  greatest  positive  value  of  y  will  correspond  to  the  smallest 
of  X ;    hence,   designating  the  superior  limit  of  the   positive    roots 

of  the  equation    1"  =  0  by  L,  we  shall  have   —  =  L\   the  inferior 

j-j 

limit  of  the  positive  roots  of  the  given  equation. 

2d.  If  in  the  equation  X=0,  we  make  a:  =  —  y,  which  gives 
the  transformed  equation  Y^i  0,  it  is  clear  that  the  positive  roots 
of  this  new  equation,  taken  with  the  sign  — ,  will  give  the  nega- 
tive roots  of  the  given  equation ;  therefore,  determining,  by  thf 
known  methods,  the  superior  limit  L  of  the  positive  roots  of  ili'' 
equation  y=:0,  we  shall  have  — i  ^  i^',  the  superior  limit  (nu- 
merically) of  the  negative  roots  of  the  proposed  equation. 


344  ELEMENTS    OF    ALGEBRA.  [CHAP.   XI. 

3d.  Finally,  if  we  replace  x,  in   the   given   equation,  by , 

y 

and  find  the   superior  limit  L  of  the  transformed  equation    Y  =  0 

then,   L'^'  = =-   will  be   the   inferior  limit   (numerically)  of  the 

negative  roots  of  the  given  equation. 

Consequences  deduced  from  the  preceding  Principles. 

First. 

320.  Every  equation  in  which  there  are  no  variations  in  the  signs, 
that  is,  in  which  all  the  terms  are  positive,  must  have  all  of  its  real 
roots  negative;  for,  every  positive  number  substituted  for  x,  will 
render  the  first  member  essentially  positive. 

Second. 

321.  Every  complete  equation,  having  its  terms  alternately  posi- 
tive and  negative,  must  have  its  real  roots  all  positive  ;  for,  every 
negative  number  substituted  for  x  in  the  proposed  equation,  would 
render  all  the  terms  positive,  if  the  equation  was  of  an  even  de- 
gree, and  all  of  them  negative,  if  it  were  of  an  odd  degree.  Hence. 
their  sum  could  not  be  equal  to   zero  in  either   case. 

This  principle  is  also  true  for  every  incomplete  equation,  in  which 
there  results,  hy  substituting  —  y  for  x,  an  equation  liaving  afl 
its  terms  affected  with  the  same  sign. 

Third. 

322.  Every  equation  of  an  odd  degree,  the  co-efficients  of  which 
are  real,  has  at  least  one  real  root  affected  with  a  sign  contrary  to 
that  of  its  last  term. 

For,  let 

^m  ^  p^m-\  -{....    Tx  ±    U  =  0, 

be   the   proposed   equation  ;    and  first  consider  the   case  in  which 
the  last  term  is  negative. 

By  making  a;  =  0,  the  first  member  becomes  —  U.  But  by 
giving  a  value  to  a:  equal  to  the  greatest  co-efficient  plus  unity, 
or  {K -\-  1))  the  first  term  x^  will  become  greater  than  the  arith- 
metical sum  of  all  the  others  (Art.  315),  the  result  of  this  sub- 
stitution will   therefore   be  positive ;   hence,   there  is   at    least    one 


CHAP.   Xr.J       CONSEQUENCES    OF    PRECEDING    PRINCIPLES.  345 

real  root  comprehended  between  0  and  K -{-  \,  Avhich  root  is  posi 
live,  and  consequently  alFected  with  a  sign  contrary  to  that  of  tlit- 
last  term  (31]). 

Suppose   now,  that  the  last  term   is  positive. 

Making   a;  =  0,   we  obtain    +  U  for  the  result ;    but  by  putting 

—  (iiL  +  1)  in  place  of  x,  we  shall  obtain  a  negative  result,  since 
the  first  term  becomes  negative  by  this  substitution ;  hence,  the 
equation  has   at  least  one   real  root   comprehended  between  0  and 

—  {K -\-  1),  which  is  negative,  or  affected  with  a  sign  contrary  to 
that  of  the  last  term. 

Fourth. 

323.  Every  equation  of  an  even  degree,  which  involves  only 
real  co-efficients  and  of  wJiich  the  last  term  is  negative,  has  at 
least  two  real  roots,  one  positive  and  the  other  negative. 

For,  let   —  Z7  be  the  last  term  ;    making   a?  ==  0,    there   results 

—  U.  Now  substitute  either  K+l,  or  —  {K -\- \),  K  being  the 
greatest  co-efEcient  in  the  equation.  As  in  is  an  even  number,  the 
first  term  x^  will  remain  positive  ;  besides,  by  these  substitutions, 
it  becomes  greater  than  the  sum  of  all  the  others  ;  therefore,  the 
results  obtained  by  these  substitutions  are  both  positive,  or  afl^ected 
with  a  sign  contrary  to  that  given  by  the  hypothesis  x=0;  hence, 
the  equation  has  at  least  two  real  roots,  one  positive,  and  compre- 
hended between  0  and  K  -\-\,  the  other  negative,  and  compre- 
hended between  0  and   —  (if  +  1),  (311). 

Fifth. 

324.  If  an  equation,  involving  only  real  co-efficients,  contains  im- 
aginary roots,  the  number  of  such  roots  must  be  even. 

For,  conceive  that  the  first  member  has  been  divided  by  all  the 
simple  factors  corresponding  to  the  real  roots  ;  the  co-efiicients 
of  the  quotient  will  be  real  (Art.  278) ;  and  the  quotient  must  also 
be  of  an  even  degree ;  for,  if  it  was  uneven,  by  placing  it  equal 
to  zero,  we  should  obtain  an  equation  that  would  contain  at  least 
one  real  root  (322) ;  hence,  the  imaginary  roots  must  enter  by  pairs. 

Remark. — 325.  There  is  a  property  of  the  above  polynomial 
quotient  M'^hich  belongs  exclusively  to  equations  containing  only 
imaginary  roots;  viz.,  every  such  equation  ahoays  remains  ■posi- 
tive for  any  real  value  substituted  for  x. 


346  ELEMENTS    OF     ALGEBRA.  [CHAP.   XI 

For,  by  substituting  for  x,  K-\-  I,  the  greatest  co-efficient  plus 
unity,  we  could  always  obtain  a  positive  result ;  hence,  if  the 
polynomial  could  become  negative,  it  would  follow  that  when 
placed  equal  to  zero,  there  would  be  at  least  one  real  root  com- 
prehended between  K  -\-  I  and  the  number  which  would  give  a 
negative  result  (Art.  311). 

It  also  follows,  that  the  last  term  of  this  polynomial  must  be 
positive,  otherwise  a;  =  0  would  give  a  negative  result. 

Sixth. 

326.  When  the  last  term  of  an  equation  is  positive,  the  number 
of  its  real  positive  roots  is  even ;  and  when  it  is  negative,  the  w.m- 
her  of  such  roots  is  uneven. 

For,  first  suppose  that  the  last  term  is  -f-  U,  or  positive.  Since 
by  making  a;  =  0,  there  will  result  +  U,  and  by  making  x=K-i-  1, 
the  result  will  also  be  positive,  it  follows  that  0  and  K  +  1  gi^'^ 
two  results  affected  with  the  same  sign,  and  consequently  (Art. 
313),  the  number  of  real  roots,  if  any,  comprehended  between  them, 
is   even. 

When  the  last  term  is  —  U,  then  0  and  K  -{-  1,  give  two  re- 
sults affected  with  contrary  sigTis,  and  consequently  comprehend 
either  a  single  root,  or  an  odd  number  of  them. 

The  reciprocal  of  this  proposition  is  evident. 

Descartes''  Rule. 

327.  An  equation  of  any  degree  rvhatever,  cannot  have  a  greater 
numl^er  of  positive  roots  tha?i  there  are  variations  in  the  signs  of  its 
terms,  nor  a  greater  number  of  negative  roots  than  there  are  per- 
manences of  these  signs. 

A  variation  is  a  change  of  sign  in  passing  along  the  terms,  and 
a  permanence  is  when  two  consecutive  terms  have   the  same  sign. 

In  the  equation  x  —  a  =  0,  there  is  one  variation,  and  one  posi- 
tive root,  X  =z  a.  x\nd  in  the  equation  x  -\-  b  =  0,  there  is  one 
permanence,  and  one  negative  root,  a?  =  —  b. 

If  these  equations  be  multiplied  together,  there  will  result  ar; 
equation  of  the  second  degree. 


a'  —  a 
+  b 


x  —  ab 


!-• 


CHAP.   XI.]  DESCARTES*    RULE.  347 

It'  a  is  less  than  b,  the  equation  will  be  of  the  first  form  (Art. 
144) ;  and  if  a  >  i  the  equation  will  be  of  the  second  form  ;  that  is, 

a  <  6     gives     a;-  +  2j;a;  —  y  =  0, 
and  a  '^  b         "         x"^  —  2px  —  ^  =  0. 

In  the  first  case,  there  is  one  permanence,  and  one  variation,  and 
in  the  second,  one  variation  and  one  permanence.  Since  in  either 
form,  one  root  is  positive  and  one  negative,  it  follows  that  there 
are  as  many  positive  roots  as  there  are  variations,  and  as  many 
negative  roots  as  there  are  permanences. 

The  proposition  will  evidently  be  demonstrated  in  a  general 
manner,  if  it  be  shown  that  the  multiplication  of  the  first  member 
by  a  factor  x  —  a,  corresponding  to  a  positive  root,  introduces  at 
least  one  variation,  and  that  the  multiplication  by  a  factor  x  -\-  a, 
corresponding  to  a  negative  root,  introduces  at  least  one  permanence. 

Take  the  equation 

^m  _|_  Ax'"-'^  db  Bx"'-'^  rb  Cx'"-^  ±   .  .  .  ±:  Tx  ±  U  =  0, 

in  which  the  signs   succeed  each   other  in  any  manner  whatever. 
By  multiplying  by  x  —  a,  we  have 


3^+'^  dzAlx'^dzB 
—  a  I       q=  Aa 


-i±  C 


a:'"-2  ±  .  .  .  ±U 
=1=  Ta 


1=0. 


The  co-efficients  which  form  the  first  horizontal  line  of  this 
product,  are  those  of  the  given  equation,  taken  with  the  same 
signs;  and  the  co-efficients  of  the  second  line  are  formed  from 
those  of  the  first,  by  multiplying  by  a,  changing  the  sigTis,  and 
advancing  each  one   place  to  the  right. 

Now,  so  long  as  each  co-efficient  of  the  upper  line  is  greater 
than  the  corresponding  one  in  the  lower,  it  will  determine  the 
sign  of  the  total  co-efficient ;  hence,  in  this  case  there  will  be, 
from  the  first  term  to  that  preceding  the  last,  inclusively,  the  same 
variations  and  the  same  permanences  as  in  the  proposed  equation  ; 
but  the  last  term  qp  Ua  having  a  sign  contrary  to  that  which  im- 
mediately precedes  it,  there  must  be  one  more  variation  than  in 
the  proposed  equation. 

When  a  co-efficient  in  the  lower  line  is  aff'ected  with  a  sign 
contrary  to  the  one  corresponding  to  it  in  the  upper,  and  is  also 
greater  than  this  last,  there  is  a  change  from  a  permanence  of  sign 
to  a  variation ;    for  the   sign   of  the  term   in  which   this   happens. 


348  ELEMENTS  OF  ALGEBRA  [CHAP.  XI. 

being  the  same  as  that  of  the  inferior  co-efRcient,  must  be  con- 
trary to  that  of  the  preceding  term,  which  has  been  supposed  to 
be  the  same  as  that  of  its  superior  co-efficient.  Hence,  each 
time  we  descend  from  the  upper  to  the  lower  line,  in  order  to 
determine  the  sign,  there  is  a  variation  which  is  not  found  in  the 
proposed  equation ;  and  if,  after  passing  into  the  lower  line,  we 
continue  in  it  throughout,  we  shall  find  for  the  remaining  terms 
the  same  variations  and  the  same  permanences  as  in  the  given 
equation,  since  the  co-efficients  of  this  line  are  all  affected  with 
signs  contrary  to  those  of  the  primitive  co-efficients.  This  sup- 
position would  therefore  give  us  one  variation  for  each  positive 
root.  But  if  we  ascend  from  the  lower  to  the  upper  line,  there 
may  be  either  a  variation  or  a  permanence.  But  even  by  sup- 
posing that  this  passage  produces  permanences  in  all  cases,  since 
the  last  term  ^  Ua  forms  a  part  of  the  lower  line,  it  will  be  ne- 
cessary to  go  once  more  from  the  upper  line  to  the  lower,  than 
from  the  lower  to  the  upper.  Hence,  the  new  equation  must  have 
at  least  one  more  variatio7i  than  the  jjroposed ;  and  it  will  be  the 
same  for  each  positive  root  introduced  into  it. 

It  may  be  demonstrated,  in  an  analogous  manner,  that  the  mul- 
tiplication by  a  factor  x  -{-  a,  corresponding  to  a  negative  root, 
would  introduce  one  permanence  more.  Hence,  in  any  equation,  the 
number  of  positive  roots  cannot  be  greater  than  the  number  of 
VARIATIONS  of  sigus,  uor  the  number  of  negative  roots  greater 
than  the  number  of  permanences. 

Consequence. 

328.  When  the  roots  of  an  equation  are  all  real,  the  number  of 
vositive  roots  is  equal  to  the  7iumber  of  variations,  and  the  number 
of  negative  roots  to  the  number  of  permanences. 

For,  let  m  denote  the  degree  of  the  equation,  n  the  number  of 
variations  of  the  signs,  p  the  number  of  permanences ;  we  shall 
have  w  =  ra  +  p.  Moreover,  let  n^  denote  the  number  of  positive 
roots,  and  p'  the  number  of  negative  roots,  we   shall  have 

m  =  71''  +  p' ; 
whence  n  -\-  p  ^  n'  -\-  p\     or,     n  —  n'  =.  p'  —  p. 

Now,  we  have  just  seen  that  n'  camiot  be  >  n,  nor  can  it  be 
ess,  since  p'  cannot  be  >  p ;  therefore  we  must  have  n'  =  n, 
and  p'  =  p. 


CHAP.   XI.]  COMMENSURABLE    ROOTS    OF    EQUATIONS.  319 

Remark. — 329.  When  an  equation  wants  some  of  its  terras,  we 
can  often  discover  the  presence  of  imaginary  roots,  by  means  of 
the  above  rule. 

For  example,   take  the  equation 

x^  -\-  px  -{-  q  ^=  0, 
p  and  g  being  essentially  positive  ;    introducing  the  term  which  is 
wanting,  by  affecting  it   with  the   co-efRcient    rt  0 :  it  becomes 
x^  ±  0  .  x'^  -{-  px  -\-  q  =  0. 

By  considering  only  the  superior  sign,  we  should  only  obtain 
permanences,  whereas  the  inferior  sign  gives  two  variations.  This 
proves  that  the  equation  has  some  imaginary  roots ;  for,  if  they 
were  all  three  real,  it  would  be  necessary  by  virtue  of  the  supe- 
rior sign,  that  they  should  be  all  negative,  and,  by  virtue  of  the 
inferior  sign,  that  two  of  them  should  be  positive  and  one  nega- 
tive, which  are  contradictor^/  results. 

We  can  conclude  nothing  from  an  equation  of  the  form 

x^  —  pa;  +  ?  =  0  ; 
for,  introducing  the  term    dz  0  .  x^,   it  becomes 
x^  ±  0  .x^  —  px  -\-  q  =  0, 
which  contains   one  permanence   and  two   variations,  whether  we 
take  the  superior  or  inferior  sign.     Therefore,  this   equation   may 
have  its  three  roots  real,  viz.,  two  positive  and  one  negative  ;   ori 
two  of  its  roots  may  be  imaginary  and  one  negative,  since  its  last 
term  is  positive  (Art.  326). 

0/  the  commensurable  Roots  of  Numerical  Equations. 

330.  Every  equation  in  which  the  co-efficients  are  whole  num- 
bers, that  of  the  first  term  being  unity,  will  have  whole  numbers 
only  for  its  commensurable  roots. 

For,  let  there  be  the  equation 

^m  J^  p^m~l  ^  Q^m-2  ^    _   _    ^  Tx  +    U  =:  0  ; 

in  which  P,  Q  .  .  .  T,  U,  are  whole  numbers,  and  suppose  that  it 
were  possible  for  one  root  to  be  a  commensurable  fraction  — 
Substituting  this  fraction  for  x,  the  equation  becomes 

fjm  fjm — 1  «m— 2  fj 

^,m    ^  ^    Im-X    ^   "<  Jm-2    -f    •   •    •    "T    -l     ^    T 


350  ELEMENTS    OF    ALGEBRA.  [CHA^.   XI. 

whence,   multiplying  both  members  by   b'"~'^,   and  transposing, 

—  =  —  Pa'^-^  —  Qa'^-^  —  ...  -  Tab""-^  —  Ub"'-K 
b 

But  the  second  member  of  this  equation  is  composed  of  a  series 
of  entire  numbers,  while  the  first  is  essentially  fractional,  for  a  and 
b  being  prime  with  each  other,  c"*  and  b  will  also  be  prime  with 
each  other  (Art.  118),  and  hence  this  equality  cannot  exist;  for,  an 
irreducible  fraction  cannot  be  equal  to  a  whole  number. 

Therefore,  it  is  impossible  for  any  commensurable  fraction  to 
satisfy  the  equation.  Now,  it  has  been  shown  (Art.  294),  that  an 
equation  containing  rational,  but  fractional  co-efficients,  can  be 
transformed  into  another  in  which  the  co-efficients  are  whole  num- 
bers, that  of  the  first  term  being  unity.  Hence  the  research  of  the 
commensurable  roots,  either  entire  or  fractional,  can  always  be  re- 
duced to  that  of  the  entire  roots. 

331.  This  being  the  case,  take  the  general  equation 

and  let  a  denote   any  entire   number,  positive   or  negative,  which 
will  verify  it. 

Since  a  is  a  root,  wc  shall  have  the  equation 
am  +  p„'"-i  4-  .  .  .  +  Ra^  Jr  Sa^  +  Ta+  U  =0  .  .  .{\). 

Replace  a  by  all  the  entire  positive  and  negative  numbers  be- 
tween 1  and  the  limit  +  L,  and  between  —  1  and  —  U^ :  those 
which  verify  the  above  equality  will  be  roots  of  the  equation.  But 
these  trials  being  long  and  troublesome,  we  will  deduce  from  equa- 
tion (1),  other  conditions  equivalent  to  this,  and  more  easily  verified. 

Transposing  all  the  terms  except  the  last,  and  dividing  by  a, 
equation  (1)  becomes 

—  =  -  a-"-!  -  Pa'"-2  —  .  .  ,  —  Ra"-  -  Sa—T  .  .  .  (2). 

a  ^   ' 

Now,  the  second  member  of  this  equation  is  an  entire  number ; 

hence    —  must  be  an  entire  number ;    therefore,  the  entire  roots  of 
a 

the  equation  are  comprised  among  the  divisors  of  the  last  term. 
Transposing    —  T"   in  equation  (2),  dividing  by  a,  ,and  making 

\-  T  =  T'',    we  have, 

a 

T 

—  =  -  a^-2  -  Pa""-^  .  .  .  —  Ra-  S  .  .  .  (3). 


CHAP.   XI.]  COMMENSURABLE    ROOTS    OF    EQUATIONS.  351 

T 

The  second  member  of  this  equation  being  entire,    — ,  that  is. 

a 

the  quotient   of  the  division  of 


-  +  r  by  a, 
a 

is  an  entire  number. 

Transposing 

the   term    —  S    and    div 

iding 

by 

a,    Ave 

have, 

by 

supposing 

T' 
a 

a 

=  —  a"'~ 

•3  _  Par-i  —  .  . 

.  -R 

.(4). 

The  second 

member 

of  this   equation 

being 

entire, 

a 

,   that 

is, 

the  quotient  of  the  division  of 

T 

\-  S   by  a 

a 

IS  an  entire  number. 

By  continuing  to  transpose  the  terms  of  the  second  member  into 
the  first,  we  shall,  after  m  —  1  transformations,  obtain  an  equa- 
tion of  the  form 

9L^-a-P. 
a 

Then,  transposing  the  term    —  P.    di^dding  by  a,  and  making 

Q'  P'  P' 

~  +  P=P',    we  have     — =— 1,    or ^1=0 

a  a  a 

This  equation,  Avhich  is  only  a  transformation  of  equation  (1), 
is  the  last  condition  which  it  is  requisite  for  the  entire  number  a 
to  satisfy,  in  order  that  it  may  be  known  to  be  a  root  of  the 
equation. 

332.  From  the  preceding  conditions  we  conclude  that,  when  an 
entire  number  a,  positive  or  negative,  is  a  root  of  the  given  equa- 
tion, the  quotient  of  the  last  term,  divided  hy  a,  is  an  entire  number. 

Adding  to  this  quotient  the  co-efficient  of  x\  the  quotient  of  this 
sum,  divided  by  a,  must  also  be  entire. 

Adding  the  co-efficient  of  a?^  to  this  last  quotient,  and  again  divi- 
ding by  a,  the  new  quotient  must  also  be  entire;    and  so  on. 


352  ELEMENTS    OF    ALGEBRA.  [CHAP.   XI 

Finally,  adding  the  co-efRcient  of  the  second  term,  that  is,  of 
x'^~^,  to  the  preceding  quotient,  the  quotient  of  this  sum  divided  by 
a,  must  be  equal  to  —  \  ;  hence,  the  result  of  the  addition  of  unity 
which  is  the  co-eficient  of  x^,  to  the  preceding  quotient,  must  be 
equal  to  0. 

Every  number  which  will  satisfy  these  conditions  will  be  a  root. 
and  those  which  do  not  satisfy  them  should  be   rejected. 

All  the  entire  roofs  may  be  determined  at  the  same  time,  as 
follows : 

After  having  determined  all  the  divisors  of  the  last  term,  write 
those  which  are  comprehended  between  the  limits  +  L  and  —  L^''  upon 
the  same  horizontal  line;  then  underneath  these  divisors  write  the 
quotients  of  (he  last  term  by  each  of  them. 

Add  the  co-eff.cient  of  x^  to  each  of  these  quotients,  and  write  the 
sums  underneath  the  quotients  which  correspond  to  them.  Then  divide 
these  sums  by  each  of  the  divisors,  and  write  the  quotients  underneath 
the  corresponding  sums,  taking  care  to  reject  the  fractional  quotients 
and  the  divisors  which  produce  them;    and  so  on. 

When  there  are  terms  wanting  in  the  proposed  equation,  their 
co-efficients,  which  are  to  be  regarded  as  equal  to  0,  must  be 
taken  into  consideration. 

EXAMPLES. 

1.  What  are  the  entire  roots  of  the  equation 

oci  —  x'^  —  13a;2  +  16a:  —  48  =  0  ? 

The  superior  limit  of  the  positive  roots  of  this  equation  (Art 
317),  is  13+  1  =  14.  The  co-efficient  48  need  not  be  considered, 
since  the  last  two  terms  can  be  put  under  the  form  16  (a;  —  3); 
hence,  when   «  >  3,   this  part  is  essentially  positive. 

The  superior  limit  of  the  negative   roots  (Art.  319),  is 

—  (1  +  V48),     or     -  8. 

Therefore,  the  divisors  of  the  last  term  which  may  give  roots, 
are  1,  2,  3,  4,  6,  8,  12;  moreover,  neither  +  1,  nor  —  1,  will 
satisfy  the  equation,  because  the  co-efficient  —  48  is  itself  greater 
than  the  sum  of  all  the  others  :  we  should  therefore  try  only  the 
positive  divisors  from  2  to  12,  and  the  negative  divisors  from  —  2 
to    —  6  inclusively. 


CHAP. 

XI.] 

COMMENSURABLE    ROOTS    OF    EQUATIONS. 

353 

By 

observing  the  rule 

given  above,  Ave  have 

12, 

8, 

6,          4, 

3,          2,   —    2, 

-    3, 

-    4, 

-    6 

-    4, 

— 

6, 

-8 

-12, 

—  16,   —24, 

+  24, 

+  16, 

+  12, 

+     8 

+  12, 

+  10, 

+  8. 

+    4, 

0,-8 

+  40, 

+  32, 

+  28, 

+  24 

+     1, 

•■ 

+    1, 

0,-4 

—  20, 

-    7, 

—    4 

-12, 

.. 

-12, 

—  13,   —  17 

-33, 

-20, 

—  17 

-    1, 

-    3, 

+     5, 

3, 

,   -    4, 
,   -    1, 

.., 

+    4, 
-    1, 

•• 

The  frst  line  contains  the  divisors,  the  second  contains  the 
quotients  arising  from  the  division  of  the  last  term  —  48,  by  each 
of  the  divisors.  The  third  line  contains  these  quotients  augmented 
by  the  co-elTicient  +16,  and  the  fourth  the  quotients  of  these 
sums  by  each  of  the  divisors  ;  this  second  condition  excludes  the 
divisors   +8,  +6,  and  —  3. 

The  fifth  is  the  preceding  line  of  quotients,  augmented  by  the 
co-efficient  —  13,  and  the  sixth  is  the  quotients  of  these  sums  by 
each  of  the  divisors ;  this  third  condition  .  excludes  the  divisors 
3,  2,   —  2,  and  —  6. 

Finally,  the  seventh  is  the  third  line  of  quotients,  augmented  by 
the  co-efficient  —  1,  and  the  eighth  is  the  quotients  of  these  suras 
by  each  of  the  divisors.  The  divisors  +  4  and  —  4  are  the  only 
ones  which  give  —  1  ;  hence,  +  4  and  —  4  are  the  only  entire 
roots  of  the  equation. 

In  fact,  if  we  divide 

a;*  —  a:3  —  13a'2  +  \Q,x  —  48, 

by  the  product    (a?  —  A)  (x  -\-  4),    or    x"^  —  16,  the  quotient  will  be 
at^  —  a;  +  3,    which  placed  equal  to  zero,  gives 

^  =  -1^1/317; 

2  2^  ' 


therefore,  the  four  roots  are 
4      —  4 


1  1      /- 

-  4 V- 

2  2^ 


11      and 


2         2    ^ 


2.  What  are  the  entire  roots  of  the  equation 

a;4  _  5a;3  4-  25x  — 21  =0? 
23 


354  ELEMENTS    OF    ALGEBRA.  [CHAP.   XT. 

3.  What  are  the  entire  roots  of  the  equation 

15a:5  _  19^4  _|_  6a;3  _i_  15^,2  ^jg^  +  6  =  0? 

4.  What  are  the  entire  roots  of  the  equation 

9x^  +  30a;5  +  22x4  _f_  lOx^  +  17x2  —  20a;  +  4  =  0. 

Sturms'   Theorem. 

333.  The  object  of  this  theorem  is  to  explain  a  method  of  de- 
termining the  number  and  places  of  the  real  roots  of  equations  in- 
volving but  one  unknown  quantity.     Let 

X=0....(1), 

represent  an  equation  containing  the  single  unknown  quantity  x\ 
X  being  a  polynomial  of  the  m"'  degree  with  respect  to  x,  the 
co-efficients  of  which  are  all  real.  If  this  equation  should  have 
equal  roots,  they  may  be  found  and  divided  out  as  in  Art.  302, 
and  the  following  reasoning  be  applied  to  the  equation  which  would 
result.     We  will  therefore  suppose  X  =  0  to  have  no  equal  roots. 

334.  Let  us  denote  the  first-derived  polynomial  of  X  by  X,,  and 
then  apply  to  X  and  X^  a  process  similar  to  that  for  finding  their 
greatest  common  divisor,  differing  only  in  this  respect,  that  instead 
of  using  the  successive  remainders  as  at  first  obtained,  we  change 
their  signs,  and  take  care  also,  in  preparing  for  the  division,  neither 
to   introduce  nor   reject   any  factor  except  a  positive   one. 

If  we   denote  the  seA'-eral  remainders,  in  order,  after  their  signs 
have  been  changed,  by  X^,  X^  .  .  .  X^,  which  are  read  X  second, 
X  third,  &c.,  and  denote   the   corresponding  quotients  by  Qj,   Qj 
.  .  Qr-i>  we  may  then  form  the  equations 

Z=X,Q,-X,  ....(2), 
X^  =  -A2Q2  —  x^ 


Xn-\   —  X^Q^  A, 


n+1 


Ar_<] 


(3). 


iSmce  by  hypothesis,  JT  =  0  has  no  equal  roots,  no  common  di- 
visor can  exist  between  X  and  X^  (Art.  300).  The  last  remainder 
—  Xj.,  will  therefore  he  different  from  zero,  and  independent  of  a;. 


CHAP,  XI.]  STURMS'    THEOREM.  355 

335.  Now,  let  us  suppose  that  a  number  p  has  been  substituted 
for  X  in  each  of  the  expressions  X,  J^j,  X^  .  .  .  X^_i ;  and  that 
the  signs  of  the  results,  together  with  the  sign  of  X^,  are  arranged 
ill  a  line  one  after  the  other :  also  that  another  number  q,  greater 
than  p,  has  been  substituted  for  x,  and  the  signs  of  the  resuUa 
arranged  in  like  manner. 

Then  will  the  number  of  variations  in  the  signs  of  the  first  ar- 
rangement, diminished  by  the  number  of  variations  in  those  of  the 
second,  denote  the  exact  number  of  real  roots  comprised  between  p 
and  q. 

336.  The  demonstration  of  this  truth  mainly  depends  upon  the 
four  following  properties  of  the  expressions  X,  X^  .  .  .  X„,  &c. 

I.  Let  a  be  a  root  of  the  equation  X  =  0.  If  we  substitute 
a  -{-  u  for  X,  and  designate  by  A  what  X  becomes,  and  denote  the 
derived  polynomials  by  A\  A^^,  A''\  &c.;  we  shall  have  (Art.  299), 

A  +   A'U   +  — -  7^2 +   «"•. 

But  since  by  hypothesis,  a  is  a  root  of  the  equation  X  =  0,  we 
have  A  =  0,  and  hence  the  above  expression  becomes 
A'^  A"' 

« (A^  +  —  «  4-  2-3-  "'••••+  «"'-0 ; 

in  wliich  A'  is  not  zero,  since  the  equation  j^  =  0  is  supposed 
not  to  contain  equal  roots.  Now  we  say,  that  u  can  be  jnade  so 
small,  that  the  sign  of  the  quantity  within  the  parenthesis  shall  be 
the  same  as  that  of  its  first  term. 

We  attain  this  object,  by  finding  for  u  a  value  which  shall  ren- 
der, numerically. 


A' 

> 

2    " 

+ 

2.3 

ti2 

+    &C 

/  ■ 

>>  V 

/A'' 

4- 

^/// 

7/. 

4-  &c 

that  is, 

a  condition  which  will  always  be  fulfilled  (Art.  310),  when 

A'        ^      . 
M  =     or     <  — ,  K  being  the  greatest  co-efFicient  of  u. 

K+A!  ° 

II.  If  any  number  be  substituted  for  x  in  these  expressions,  it  is 
impossible  that  any  two  consecutive  ones  can  become  zero  at  the  same 
time. 


356  ELEMENTS    OF    ALGEBRA.  [CHAP.   Xi 

For,  let  X„_„  X^,  -^n+u  be  any  three  consecutive  expressions, 
Then   among  equations  (3),  we  shall  lind 

Xn-i  —  X^Qn  —  -^n+l  •   •   •   •  (4), 

from  which  it  appears  that,  if  X„_i  and  X^  should  both  become  0 
for  a  value  of  x,  X,^J^^  would  be  0  for  the  same  value ;  and  since 
the  equation  which  follows  (4)  must  be 

we  shall  have  X„^2  =  0  for  the  same  value,  and  so  on  until  we 
should  find  X^  =  0,  which  cannot  be  ;  hence,  X,j_i  and  X„  can- 
not both  become   0   for  the   same   value  of  x. 

III.  By  an  examination  of  equation  (4),  Ave  see  that  if  X„  be- 
comes 0  for  a  value  of  x,  A''„_i  and  A'',,^,  must  have  contrary 
signs  ;  that  is,  if  any  one  of  the  expressions  is  reduced  to  0  by 
the  substitution  of  a  value  for  x,  the  preceding  and  following  ones 
will  have  contrary  signs  for  the  same  value. 

IV.  Let  us  substitute  a  -\-  u  for  x  in  the  expressions  X  and  X^, 
and  designate  by  U  and  ZJ,  what  they  respectively  become  under 
this  supposition.     Then  (Art.  297),  we  have 

U  =  A   -\-  A'u  +  A"  —  ^  &c. 

\.  ;    -  ■  .  .  (5), 

U,  =  A,-{-A\u-\-A'\^+  &c. 

in  which  A,  A' ,  A",  &,c.,  are  the  results  obtained  by  the  sunsti 
tution  of  a  for  x,  in  X  and  its  derived  polynomials  ;  and  Ai  A\  &c., 
are  similar  results  derived  from  X^.  If  now,  a  be  a  root  of  the 
proposed  equation  X  =  0,  then  A  =  0,  and  since  A'  and  A^  are 
each  derived  from  A""!,  by  the  substitution  of  a  for  a-,  we  have 
A'  =  A^,  and  equations  (5)  become 

U  =A'u  +  A-^+Slc.}  ^„. 

2  S (6). 

Uy=    A'+  A'^w    +  &c.  ) 

Now,  the  arbitrary  quantity  u  may  be  taken  so  small  that  when 

added  to  a,  it  will  but  insensibly  increase  it,  and  when  subtracted 

from  a,  it  will  but  insensibly  diminish  it;  in  which  cases,  the  signs 

of  the  values  of   \J  and    V^  will   depend   upon   the   signs   of  their 

first  terms ;    that  is,  they  will  be  alike  w^hen  u  is  positive  or  when 

c  +  "   is  substituted  for  x,  and  unlike  when  u  is  negative  or  whsn 


CHAP.   XI.]  STURMS'    THEOREM.  357 

a  —  u  is  substituted  for  x.  Hence,  if  a  number  insensihly  less  than 
one  of  the  real  roots  of  X  =:  0  be  substituted  for  x  in  X  and  X„ 
the  results  will  have  contrary  signs,  and  if  a  number  insensibly  greater 
than  this  root  be  substituted,  the  results  will  have  the  same  sign. 

337.  Now,  let  any  number  as  k,  algebraically  less,  that  is,  nearer 
equal  to   —  oo,  than  any  of  the  real  roots  of  the  several  equations 

X=0,     X,  =  0  .  .  .  X^_,  =  0, 

be  substituted  for  x  ia  them,  and  the  signs  of  the  several  results 
arranged  in  order ;  then,  let  x  be  increased  by  insensible  degrees, 
until  it  becomes  equal  to  h  the  least  of  all  the  roots  of  the  equa- 
tions. As  there  is  no  root  of  either  of  the  equations  between  k  and 
h,  none  of  the  signs  can  change  while  x  is  less  than  h  (Art.  311), 
and  the  number  of  variations  and  permanences  in  the  several  sets 
of  results,  will  remain  the  same  as  in  those  obtained  by  the  first 
substitution. 

When  X  becomes  equal  to  h,  one  or  viore  of  the  expressions  X, 
X,,  &c.,  will  reduce  to  0.  Suppose  X^  becomes  0.  Then,  as  by 
the  second  and  third  properties  above  explained,  neither  -X„_,  nor 
Xnj^y  can  become  0  at  the  same  time,  but  must  have  contrary 
signs,  it  follows  that  in  passing  from  one  to  the  other  (omitting 
Xn  =  0),  there  will  be  07ie  and  only  one  variation  •  and  since  their 
signs  have  not  changed,  one  must  be  the  same  as,  and  the  other 
contrary  to,  that  of  X„,  both  before  and  after  it  becomes  0  ;  hence, 
in  passing  over  the  three,  either  just  before  X^^  becomes  0  or  just 
after,  there  is  one  and  only  one  variation.  Therefore,  the  reduc- 
tion of  X„  to  0  neither  increases  nor  diminishes  the  number  of 
variations ;  and  this  will  evidently  be  the  case,  although  several 
of  the  expressions  X^,  X^,  &c.,  should  become  0  at  the  same  time. 

If  X  =^  h  should  reduce  X  to  0,  then  h  is  the  least  real  root  of 
the  proposed  equation,  which  root  we  denote  by  a ;  and  since  by 
the  fourth  property,  just  before  x  becomes  equal  to  a,  the  signs 
of  X  and  Xi  are  contrary,  giving  a  variation,  and  just  after  pass- 
ing it  (before  x  becomes  equal  to  a  root  of  X,  =  0),  the  signs  are 
the  same,  giving  a  permanence  instead,  it  follows  that  in  passing 
this  root  a  variation  is  lost.  In  the  same  way,  increasing  x  by 
insensible  degrees  from  x  ^  a  -\-  u  until  we  reach  the  root  of 
AT  =  0  next  in  order,  it  is  plain  that  no  variation  will  be  lost  or 
gained  in  passing  any  of  the  roots  of  the  other  equations,  but  that 


358  ELEMENTS    OF    ALGEBRA.  [CHAP.   XI 

in  passing  this  root,  for  the  same  reason  as  before,  another  varia- 
tion will  be  lost,  and  so  on  for  each  real  root  between  k  and  the 
number  last  substituted,  as  g,  a  variation  will  be  lost  until  x  has 
been  increased  beyond  the  greatest  real  root,  when  no  more  can  he 
hst  or  gained.  Hence,  the  excess  of  the  number  of  variations  ob- 
tained by  the  substitution  of  k  over  those  obtained  by  the  substi- 
tution of  g,  will  be  equal  to  the  number  of  real  roots  comprised 
between  k  and  g. 

It  is  evident  that  the  same  course  of  reasoning  will  apply  when 
we  commence  with  any  number  p,  whether  less  than  all  the  roots 
or  not,  and  gradually  increase  x  until  it  equals  any  other  number  q. 
The  fact  enunciated  in   Art.  335  is  therefore   established. 

338.  In  seeking  the  number  of  roots  comprised  between  p  and  q, 
should  either  p  or  q  reduce  any  of  the  expressions  Xi,  X^,  &c., 
to  0,  the  result  will  not  be  affected  by  their  omission,  since  the 
number  of  variations  will  be  the  same. 

Should  p  reduce  X  to  0,  then  p  is  a  root,  but  not  one  of  those 
sought ;  and  as  the  substitution  oi  p  -\-  u  will  give  X  and  X,  tlio 
same  sign,  the  number  of  variations  to  be  counted  will  not  be 
affected  by  the  omission  of  X  =  0. 

Should  q  reduce  X  to  0,  then  q  is  also  a  root ;  and  as  the 
substitution  of  5'  —  u  will  give  X  and  X,  contrary  signs,  one  varia- 
tion must  be  counted  in  passing  from  X  to  X^. 

339.  If  in  the  application  of  the  preceding  principles,  we  ob- 
serve that  any  one  of  the  expressions  X,,  X2  .  .  .  &c.,  X,^  for  in- 
stance, will  preserve  the  same  sign  for  all  values  of  x  in  passing 
from  p  to  q,  inclusive,  it  will  be  unnecessary  to  use  the  succeed- 
ing expressions,  or  even  to  deduce  them.  For,  as  X^  preserves 
the  same  sign  during  the  successive  substitutions,  it  is  plain  that 
the  same  number  of  variations  will  be  lost  among  the  expressions 
X,  Xi,  &c.  .  .  .  ending  with  X„  as  among  all  including  X^.  When- 
ever then,  in  the  course  of  the  division,  it  is  found  that  by  placing 
any  of  the  remainders  equal  to  0,  an  equation  is  obtained  with 
imaginary  roots  only  (Art.  325),  it  will  be  useless  to  obtain  any 
of  the  succeeding  remainders.  This  principle  will  be  found  very 
useful  in  the  solution  of  numerical  examples. 

340.  As  all  the  real  roots  of  the  proposed  equation  are  neces- 
sarily included  between   —  oo  and  -f  oo,  we  may,  by  ascertaining 


CHAP.  XI.]  STURMS'    THEOREM.  359 

the  number  of  variations  lost  by  the  substitution  of  these,  in  suc- 
cession, in  the  expressions  X,  X^  .  .  .  X„,  .  .  &c.,  readily  determine 
the  total  number  of  such  roots.  It  should  be  observed,  that  it  will 
be  only  necessary  to  make  these  substitutions  in  the  first  terms 
of  each  of  the  expressions,  as  in  this  case  the  sign  of  the  term 
will  determine  that  of  the   entire   expression  (Art.  315). 

341.  Having  thus  obtained  the  total  number  of  real  roots,  we 
may  ascertain  their  places  by  substituting  for  x,  in  succession, 
the  values  0,  1,  2,  3,  &c.,  until  we  find  an  entire  number  which 
gives  the  same  number  of  variations  as  +03.  This  will  be  the 
smallest  superior  limit  of  the  positive  roots  in  entire  numbers. 

Then  substitute  0,  —  1,  —  2,  &c.,  until  a  negative  number  is 
obtained  which  gives  the  same  number  of  variations  as  —  oo  This 
will  be,  numerically,  the  smallest  superior  limit  of  the  negative 
roots  in  entire  numbers.  Now,  by  commencing  with  this  limit  and 
observing  the  number  of  variations  lost  in  passing  from  each  num- 
ber to  the  next  in  order,  we  shall  discover  how  many  roots  are 
included  between  each  two  of  the  consecutive  numbers  used,  and 
thus,  of  course,  know  the  entire  part  of  each  root.  The  decimal 
part  may  then  be  sought  by  some  of  the  known  methods  of  ap- 
proximation. 

EXAMPLES. 

1.  Let  8x3  —  6a;  —  1  =  0  =  Z. 

The  first-derived   polynomial   (Art.   297),   is 
24*2  _  Q^ 

and  since  we  may  omit  the  positive  factor  6,  without  affecting  the 
sign,  we  may  write 

4a:2  -  1  =:  X,. 

Dividing  X  by  X,,  we  obtain  for  the  first  remainder,  —  4x  —  1. 
Changing  its  sign,   we  have 

4*  +  1  =  X^. 

Multiplying  X^  by  the  positive  number  4,  and  then  dividing  by 
X2,  we  obtain  the  second  remainder  —  3  ;  and  by  changing  its 
sign 

+  3  =  Z3. 

The  expressions  to  be  used   are  then 
X=8x^  ~6x-  1,     Z,  =  4a:2  -  1,     X^  =  4x  +1,     X3  =  +  3 


360  ELEMENTS    OF    ALGEBRA.  TOHAP.  XI. 

Sul)stituting  —  00  and  then  +  oo,  we  obtain  the  two  following 
arrangements  of  signs : 

—  4"  —  + 3  variations, 

+  +  +  +••  .0 

There  are  then  three  real  roots. 

If  now,  in  the  same  expressions  we  substitute  0  and  +  1, 
and  then  0  and  —  1,  for  x,  we  shall  obtain  the  three  following 
arrangements : 

For  X  =  -{-  1  +  +  +  +  0  variations, 

x=        0  \-  +  1 

x=  -I  -  -\-  -  i-  3 

As  a;  =  +  1  gives  the  same  number  of  variations  as  +  °o,  and 
X  =  —1  gives  the  same  as  —  oo,  +1  and  —  1  are  the  smallest 
limits  in  entire  numbers.  In  passing  from  —  1  to  0,  two  variations 
are  lost,  and  in  passing  from  0  to  +  ],  one  variation  is  lost;  hence, 
there  are  two  negative  roots  between  —  1  and  0,  and  one  positive 
root  between  0  and  +  1. 

2.  Let  2x*  —  13x2  _|_  lOa:  -19  =  0. 

If  we  deduce  X,  X,,  and  Xg,  Ave  have  the  three  expressions 
X  =    2x*  —  13x2  +  lOx  -  19, 
X,  =    4x3  _  13^    +    5^ 
Xg  =  13x2  _  15-^    ^  38. 

If  we  place  X^  =  0,  we  shall  find  that  both  of  the  roots  of 
the  resulting  equation  are  imaginary  ;  hence,  X^  will  be  positive 
for  all  values  of  x  (Art.  325).  It  is  then  useless  to  seek  for  X, 
and  X4. 

By  the  substitution  of  —  00  and  +  co  in  X,  X,,  and  Xj,  we 
obtain  for  the  first,  two  variations,  and  for  the  second  none ;  hence, 
there  are  two  real  and  two  imaginary  roots  in  the  proposed  equation. 

3.  Let  x3  —  5x2  +  8x  —  1  =  0. 

4.  X*  —    x3  —  3x2  +  x2  —  a:  -  3  =  0. 

5.  x5  _  2a;3  +  1  =  0. 
Discuss  each  of  the  above  equations. 


CHAP.   XI.]  RESOLUTION    OF    CUBIC    EQUATIONS.      •  361 

Young's   Method  of  resolcing   Cubic  Equations. 

342.  Every  numerical  equation  of  the  third  degree  may  be  re- 
duced to  the  form 

x^ -{- Ix^  +  ex  -  N         (1), 

in  which  b,  c,  and  N,  are  known  numbers. 

Since  this  equation  will  have  at  least  one  real  root  (Art.  277), 
let  us  find,  either  by  Sturms'  theorem  or  by  trial,  two  consecutive 
numbers,  either  integral  or  decimal,  which  being  substituted  for  ar, 
will  give  results  with  different  signs.  We  then  know  that  one  of 
the  values  of  x  will  lie  between  them  (Art.  311),  and  consequently, 
that  the  smallest  number  will  be  the  first  figure  of  the  required 
root. 

Let  us  designate  this  figure  by  r.  Now,  if  we  neglect  the  re- 
maining figures  of  the  root,  and  regard  r  as  the  approximate  value 
of  X,  we  shall  have 

N 

r^  -\-  hr^  -{-  cr  ^:z  N ;    whence,    r  = . 

r^  -\-  or  -\-  c 

Having  found  r,  denote  the  remaining  part  of  the  root  by  y ;  then 

X  ^=  r  -\-  y. 

Substituting  this  value  of  x  in  the  given   equation,    we   have 

^3   __   y.3   _[_  3j-2y   _|_    3r?/2   _[_  y3   \ 

bx"^  =  br^  -f  2bry  +  hf  >  =  iV 

ex   z=  cr  -\-  cy  ) 

and  by  adding  and   arranging  with  reference  to  y, 

y3  +  (3r  +  b)y'^  +  (3r2  -\.  2br  +  c)  y  +  (r^  +  Z-r2  +  cr)  =  N. 
But  we   may  simplify  the  form  of  this  equation,  by  making 

Z/'  =  3r  +  b,       c'  =  3r2  +  2Z/r  +  c,       N'  =  N  —  {r^  +  b?-^-  +  cr) ; 

which  will  give       if  +  b^^  +  c^y  =  N'         (2). 

The  form  of  this  equation  is  entirely  similar  to  that  of  the  given 
equation  ;  and  if  we  denote  by  s  the  first  figure  of  y  and  make 
tne   same  supposition  as   before,  w^e   shall  have 

wlience,  s  = . 

*2  ^  b's  -i-  </ 


362  ELEMENTS    OF    ALGEBRA.  ,  [(JHAP.  XI. 

Supposing  the  value  of  s  found ;  denote  by  z  the  remaining 
figures  of  the  root :    then 

y    =:S  +  z, 
and  y3  =  ^-^  +  ^s-z  +  3^2^  +  z^y 

b'lf  =  b's^  +  2//SZ  +  b'z^, 
c'y    =  c's  -\-  (/z. 
By  adding  and  arranging  with  reference   to  2, 
23  +  (3.y  +  V)  s2  +  (3^2  _^  2/j'5  +  cQ  0  +  (^3  +  ^V  +  (^s)  =  iV^'. 
But,  by  making         h'^  =  3^  +  h\ 

d'  —  3^2  +  2h's  +  c', 

the  equation  becomes 

23  4-  ///^s  +  c''z  =  N''         (3). 

If  we  denote  by  t  the  first  figure  of  z,  and  make  the  same 
supposition  as  before,  we   shall   have 

t3  _|_  y'l^  +  c''t  =  N'' ; 

If  we  designate  by  u  the  first  figure  after  t,  we  should  find, 
by  a  process  similar  to  the  above, 

"  "  ?i2  +  5'^^M  +  c^^' ' 

and  in  a  similar  manner  we  may  find  the  algebraic  expression  for 
any  succeeding  figure  of  the  root. 

343.  It  is  now  required  to  put  these  algebraic  expressions  under 
such  a  form  as  will  indicate  the  best  practical  rule  for  performing 
the  arithmetical  operations.  For  this  purpose,  let  us  bring  the  for- 
mulas together.     We  have 

N  \  N 


r^  +  br  -\-  c 

si  +  b's  +  C' 

~  ;?  (5  +  3r  +  Z>)  +  3r2  +  2ir  +  c ' 

fl  +  b"t  +  c'' 

""  f  [«  +  3  (r  +  5)  +  i]  +  3^2  +  2b's  +  c" ' 

y2  _^  5///y  _|_  g///  „  [y   4.  3  (r  _f.   5  ^  f )    _}.  Jj   _[_  3^2  4.  2b" t  4-   C' 

&c.,  &c.,  &c 


CHAP.  XI.]  RESOLUTION    OF    CUBIC    EQUATIONS.  36.S 

The  value  of  r  being  found,  and  c  a  known  number,  tlie  de- 
nominator 7-(r4-5)H-c  will  be  known.  This  forms  the  first  di- 
visor, and  dividing  N  by  it,  the  first  figure  of  the  quotient  will  be 
r,  as  before  found.  Multiplying  the  divisor  by  r  and  subtracting 
the  product  from   N,  we  obtain   N' ,  the  second  dividend. 

It  will  be  seen  that  the  three  right-hand  terms  in  each  denomin- 
ator, are  formed  from  the  preceding  figure  of  the  root.  These 
make  trial  divisors  for   each  figure  of  the   root  after  the  first. 

Having  found  N\  we  form  its  trial  divisor  and  then  see  how 
often  it  is  contained  in  N^,  which  gives  s.  We  then  form  the 
complete  divisor  which  we  multiply  by  s,  and  subtract  the  result 
from  N'',  which  gives  N^''.  We  then  form  its  trial  divisor,  find  the 
figure  t,  after  which  we  find  the  complete  divisor  for  t,  and  then 
multiply  it  by  the  quotient  figure  t  and  subtract  the  result  from  iNP'', 
giving  N'"''';  and  similarly  for  all  the  following  figures  of  the  root. 

344.  By  examining  the  table  of  Example  I,  on  the  next  page, 
and  comparing  it  with  the  formulas,  we  see,  that  if  under  any  divi- 
sor we  write  the  square  of  the  figure  of  the  root  which  the  divisor 
determines,  and  then  add  it  to  the  two  numbers  directly  above,  their 
sum  will  be  the  trial  divisor  for  the  next  figure  of  the  root.  Hence, 
we  have  the  following 

RULE. 

I.  Write  doicn  c,  the  co-effcient  of  x,  and  on  the  same  line,  to  the 
right,  place  the  known  number  N,  and  set  in  the  quotient  the  Jirst  figure 
of  the  root  found  by  trial. 

n.  To  this  figure  of  the  root  add  b,  the  co-ejfficient  of  x-,  and 
then  multiply  the  sum  hy  the  figure  of  the  root,  and  add  the  product 
to  c,  and  the  sum  will  be  the  first  divisor,  which  is  then  to  be 
multiplied  by  tlie  quotient  figure  and  the  product  subtracted  from  N. 

HI.  Under  the  first  divisor  write  the  square  of  the  first  figure  of 
the  root,  and  then  add  it  to  the  last  two  sums,  and  the  result  will 
be  tJie  trial  divisor  for  the  next  figure  of  the  root. 

IV.  Having  found  the  next  figure  of  the  root,  add  to  it  three 
times  the  figures  of  the  root  already  found,  and  also  the  co-efficient 
t :  then  multiply  the  sum  by  the  figure  of  the  root  and  add  the  prod- 
uct to  the  trial  divisor,  and  the  sum  will  be  the  entire  divisor,  which 
must  then  be  multiplied  by  the  figure  of  the  root,  and  the  product 
subtracted  from  the  last  dividend.  The  process  for  determining  other 
figures  of  the  root  is  entirely  similar. 


364 


ELEMENTS    OF    ALGEBRA. 


[CHAP.  XI. 


+ 

05 


+ 

IN 

+ 


lO      l>1 


r3 

3 

w 

CO 

G 

II 

a 

II 

■^ 

]l 

^ 

u 

'S 

^ 
^ 

CO 

+ 

■5^ 

+ 

II 

is 

II 

•v. 

II 

^ 

II 

+ 

+ 

5: 

+ 

> 

fe: 

%> 

fe; 

•*o 

fe; 

^ 

<: 

II 

II 

II 

II 

II 

II 

in 
to 

II 

CO 

00 

CJ 

r- 

^ 

00 

rsf 

CX) 

CO 

CO 

XI 

^ 

^ 

CO 

C5 

o 

rH 

Tf 

o 

C5 

CO 

o 

^H 

'Tf 

05 

CO 

lO 

ct 

CO 

CO 

o 

CO 

C9 

T-H 

T— 1 

o 

o 

o 

w 

CN( 

=3 


a 
.9 

&     • 


jV 


CO  =  a 


H    + 


^    + 


oo  =  q 


CO      O    1  CO 


in 

in 

fN> 

r* 

-.-« 

Tf 

Tt< 

(>J 

CO 

in 

'^ 

Tf 

o 

CT5 

CO 

in 

CO 

CO 

CD 

Of) 

o 

00 

o 

00 

f^ 

en 

t> 

J> 

CO 

CO 

o 

o 

w 

o 

CO 

o 

in 

^ 

^ 

o 

w 

o 

rs> 

in 

CO 

CO 

CO 

CO 

00 

+ 

u 

(N 

+ 

(N 

^ 

II 

+ 

lO 

V 

+ 

!;. 

V 

II 

V. 

>-l 

II  . 

+  + 

+  + 1 

^  c^  ^ 

"  +  + 


+ 

+ 


> 


+ 

^  + 
+  ^ 
^  + 

CO 

+ 


+ 


+    + 


+ 


CO 

+ 


+ 

to 

+ 

CO 

+ 


.2 

n3 

TS 

r^ 

K 

CO 

^ 

CHAP.   XI.] 


RESOLUTION    OF    CUBIC    EQUATIONS. 


36') 


Remark. — The  operations  in  tlie  example  of  the  table,  are  all 
performed  according  to  the  directions  of  the  rule  ;  but  more  deci- 
mal places  have  been  used  in  the  dividends  and  divisors,  in  the 
latter  part  of  the  work,  than  were  necessary.  Had  we  admitted 
but  three  places  of  decimals  in  the  dividends,  and  rejected  all  other 
places  to  the  right,  as  fast  as  they  occurred,  we  should  still  have 
'lad  the  root  equally  true  to  at  least  four  places  of  decimals.  But 
since  the  figures  of  the  root  are  decimals,  it  follows  that  if  the  num- 
ber of  decimal  places  in  the  dividend  does  not  exceed  three,  the 
decimal  places  in  the  corresponding  divisor  should  not  exceed  two ; 
and  for  every  succeeding  figure  found  in  the  root,  one  place  may 
be  struck  ofl"  from  the  right  of  the  divisor. 

After  finding  a  certain  number  of  figures  of  the  root,  it  will 
occur  that  the  numbers  to  be  added  to  the  divisors  will  fall  among 
the  rejected  figures,  after  which  the  remaining  figures  of  the  root 
will  be  found  by  simple  division.  It  should  be  observed,  however, 
that  when  places  are  rejected  from  the  divisor,  that  whatever  would 
have  been  carried  had  the  complete  multii)lication  been  performed, 
is  still  to  be  carried  to  increase  the  last  figure  retained ;  and  when- 
ever the  left-hand  figure  of  those  rejected,  either  in  the  dividend  or 
divisor,  exceeds  4,  the  last  figure  retained  is  to  be  increased  by  1. 

The  following  is  the  last  example,  wrought  on  the  principle  of 
admitting  but  three  places  of  decimals  into  the  dividends.  The 
rejected  figures,  both  in  the  dividends  and  divisors,  are  placed  a 
little  to  the  right. 

6 
20 

26 
4 

60 
5.76 


55  .76 

.16 

61  .68 


30 

44 

62 

0 
0 

844 
004 

62 

3 

1 

892 
76325 

62 

.4 

75 

9  12. 

4257  4- 

52 

23 

9 

22 

.304 

1 

.596 

1 

.240 

088 

.356 

312 

.312 

827625 

.044 

484375 

043 

001 


366  ELEMENTS    OF    ALGEBRA.  [CHAP.   SI. 

2.  Find  one  root  of  the  equation    x^  +  x-  =  500. 
This  equation  is  the  same  as   x^  +  x-  +  Oa?  =  500  ;  hence  6  =  1 
c  =  0,   and  N  =  500. 

The   first  figure  of  the  root  found  by  trial,  is  7. 

0 
56 

56 
49 

Tel 

13  .56 


174 

.56 
36 

188 

.48 
.2381 

188 

.7181 
.0001 

188 

.9563 
.  1669 

189, 

.  123|2 

189, 

.290 
.005 

500 

[7.61727975, 

&C.,  =  X 

392 

108 
104 

.736 

3Ught 

the 
lence 
ends, 
imals 

3 

.264 

J-,   C   C  't3   CJ 

1 

.887181 

<«  2  S  "^  2 

] 

.376819 

■^       O                 q;}       O 

1 

. 323862 

C^           -.^     a     (J 

. 052957 
.  037859 

exam 
cimal 
ay  ah 
s   fron 

3  pla 

. OlSODa 
.013251 

^       cd  s 
C-,  >o  .<^  'E,  S 

.001847 

bD  '^   03   Kr 

.001704 

■a                      r^          ^ 

- 

.000143 

^•1^    ^'^ 

.000133 

Rem 
y  ret 
ividei 
ejectii 
fler  h 

.000010 

. 000009 

_o  --d  >-,  c; 

"    o 


.  1|8|9.12|9|5 

3.  F'ind  one  root  of  the  equation   x^  —  17a:-  +  54a:  =  350. 

Ans.  a-  =  14  .  954,  &c. 

4.  Find  one  root  of  the  equation    x^  +  2x'^  +  3x  =:  13089030. 

Ans.  X  =  235. 

5.  Find  one  root  of  the  equation   x^  4-  Sx^  —  23x  =  70. 

Ans.  X  =  5  .  1345,  &c. 

Remark. — In  the  preceding  solutions  only  one  root  has  been 
obtained,  yet  the  others  may  be  found  with  equal  facility,  by  find- 
ing by  trial  the  first  figure  in  each  and  then  proceeding  by  the 
rule  already  given.  There  is,  however,  a  shorter  inelhod  for  de- 
termining the  remaining  roots. 

Subtract  the  root  found,  taken  with  a  contrary  sign,  from  the  co- 
efficient of  the  second  term  of  the  given  equation,  and  denote  the 
remainder  by  a.  Divide  the  absolute  term  by  the  root  found,  and 
denote  the  quotient  by  b ;    then  will  the  roots   of  the   equation 

a;2  -f-  CT  -f  &  =  0 
be  the  two  remaining  roots  of  the  giA^en  equatiop 


CHAP.   XI.]  RESOLUTIOX    OF    TIIGHE.'J    KQUATION3.  367 


Method  of  resolving  Higher  Equations. 

345.  The  general  method  of  resolving  cubic  equations,  has  been 
explained  in  Art.  342.  We  shall  now  add  from  Young's  Algebra, 
the  method  of  resolving  equations  of  a  higher  degree.  It  has 
not  been  thought  best  to  give  the  general  investigation,  but  merely 
to  add,  for  the  solution  of  an  equation  of  any  degree,  the  follow- 
ing general 

RULE. 

I.  Transpose  the  absolute  term  to  the  second  memher  of  the  equation. 
Then,  beginning  with  the  co-efficient  of  the  first  term,  arrange  the  co- 
efficients of  the  first  member  in  a  row,  placing  the  absolute  term  to  the 
right. 

II.  Having  found  the  first  figure  of  the  root,  multiply  it  by  the  first 
co-efficient  and  add  the  product  to  the  second  co-efficient ;  then  multiply 
the  sum  by  the  same  figure  of  the  root  and  add  the  product  to  the  third 
cn-effiicient ;  and  so  on  to  the  last  co-efficient :  the  last  sum  will  be  the  first 
divisor,  which  multiply  by  the  figure  of  the  root  and  subtract  the  product 
from  the  absolute  term :   the  result  will  be  the  second  dividend. 

III.  Perform  the  same  operations  on  the  first  co-efficient  and  the  set  of 
swns  found,  as  was  performed  unth  the  co-efficients,  and  the  last  sum  xvill 
be  the  trial  divisor  for  the  second  figure  of  the  root.  Then  perform 
the  same  ojjerations  on  the  first  co-efficient  and  the  second  set  of  sums,  only 
stop  in  the  column  of  the  last  co-efficient  but  one.  Repeat  the  same  op- 
eration on  the  first  co-efficient  and  the  last  set  of  sums,  but  stop  in  the 
next  left-hand  column,  and  so  on  until  you  stop  in  the  column  of  the  sec- 
ond co-efficient. 

IV.  Then  find  from,  the  trial  divisor  the  second  figure  of  the  root, 
taking  care  that  it  be  not  too  large.  Take  this  second  figure,  and  per- 
form unth  it  on  the  first  co-efficient  and  the  last  set  of  sinus  the  same 
operations  as  were  performed  on  the  co-efficients  with  the  first  figure  of 
Oie  root;  and  the  sum  in  the  last  column  will  be  the  second  divisor,  which 
multiply  by  the  second  figure  of  the  root  and  subtract  the  product  from 
the  second  dividend. 

V.  The  next  trial  divisor,  the  next  figure  of  the  root,  and  the  true 
divisor,  arc  found  by  the  principles  already  explained,  and  the  placet^ 
of  figures  in  the  root  may  be  carried  as  far  as  necessary. 


368  ELEMENTS    OF    ALGEBRA.  [CHAP.   XI 

EXAMPLES. 

1.  Find  the  root  of  the   equation   x*  —  3x^  +  75a?  =  10000. 
Operatio7i. 


0 

-3 

75 

10000 

[9.8860027,  &c., 

,  =  y 

9 

81 

702 

6993 

9 

78 

777 

3007 

9 

162 
240 

2160 

2677. 

,5616 

18 

2937 

329 

.4384 

9 

243 
483 

409. 

952 

306 

.  1662 

27 

3346. 

952 

23 

.2722 

9 

,8 

29. 
512. 

44 
44 

434, 

.016 

23 

.2616 

36. 

3780. 

968 

106 

,8 

30, 

,08 

46 

.  110 

78 
28 

37. 

,6 

542  . 

52 

3827. 

,  07|8 

.  8 

30, 

,72 

46 

.36 

27 
T 

38. 

.4 

573  . 

,24 

3873, 

.44 

.8 

3  . 

,  14 

3 

.50 

3|9, 

.|2 

576 

.3|8 

3876 

.  9|4 

3 

579 

.  1 

•  1^ 

3 

.5 

3880 

.4 

3 

5|8|3 

Remark. — The  work,  in  the  example,  has  been  contracted  by 
omitting  or  cutting  off  decimal  places,  as  in  the  operations  for  tho 
cube  root,  and  in  equations  higher  than  the  third  degree,  the  con- 
tractions may  be  begun  after  the  first  decimal  place  of  the  root  is 
found. 

By  using  one  period  of  four  decimal  places,  the  root  has  been 
found  to  eight  places  of  figures.  With  another  period  of  Your  places, 
that  is,  by  beginning  the  contractions  later,  we  should  have  found 
four  additional  places,  or  ic  =  9  .  88600270094. 


THE  END. 


QA.     Davies  -  Elements  of  algebra:  including 

iSh  Sturms*  theorem. 

D28E 

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